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Theory of Job Search.Unemployment-Participation Tradeoff andSpatial Search with Asymmetric Changes
of the Wage Distribution
Dissertation zur Erlangung des Grades eines Doktorsder Wirtschaftswissenschaften
eingereicht an der Wirtschaftswissenschaftlichen Fakultat
der Universitat Regensburg
vorgelegt von
Alisher Aldashev
Betreuer:
Prof. Dr. Joachim Moller
Prof. Dr. Lutz Arnold
Datum der Disputation 5. Juli 2007
Theory of Job Search.
Unemployment-Participation Tradeoff and Spatial
Search with Asymmetric Changes of the Wage
Distribution
Alisher Aldashev
ZEW, Mannheim
Mannheim, November 14, 2007
Contents
1 Introduction 1
2 Review of Literature 6
2.1 Models with Constant Wages (Urn Models) . . . . . . . . . . . . . .. . . . . 8
2.1.1 Full Equilibrium Model with Employment Agency . . . . . .. . . . . 8
2.1.2 Equilibrium Unemployment Rate and Employment Duration . . . . . . 11
2.1.3 Spatial Search with Constant Wages . . . . . . . . . . . . . . . .. . . 13
2.2 Exogenous Wage Dispersion . . . . . . . . . . . . . . . . . . . . . . . . .. . 14
2.2.1 Single Wage Offer Model, Discrete Time . . . . . . . . . . . . .. . . 14
2.2.2 Multiple Wage Offer Model, Continuous Time . . . . . . . . .. . . . 16
2.3 Endogenous Wage Dispersion . . . . . . . . . . . . . . . . . . . . . . . .. . 18
2.3.1 Wage Dispersion Due to Worker Heterogeneity . . . . . . . .. . . . . 18
2.3.2 Equilibrium Wage Dispersion with Identical Workers .. . . . . . . . . 24
2.4 Conclusion and Empirical Relevance . . . . . . . . . . . . . . . . .. . . . . . 26
3 Nonstationarity in the Theory of Job Search and Withdrawals from the Labor
Market 28
3.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 28
3.2 Unemployment Participation Tradeoff . . . . . . . . . . . . . . .. . . . . . . 30
3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
i
4 Empirical Estimation of Duration Models 35
4.1 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
4.1.1 Parametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Proportional Hazard Specification and Semiparametric Estimation . . . 38
4.1.3 Competing Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.4 Nonparametric Methods: Kaplan-Meier Product-LimitEstimator . . . 40
4.2 The Kaplan-Meier Estimator and Withdrawals from the Labor Market . . . . . 41
5 Spatial Search Theory and Commuting 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
5.2 Bilocational Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 49
5.3 Maximal Acceptable Travel Cost . . . . . . . . . . . . . . . . . . . . .. . . . 54
5.4 Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
6 Empirical Estimation of a Commuting Model 58
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
6.2 Data and Descriptive Evidence . . . . . . . . . . . . . . . . . . . . . .. . . . 59
6.3 Clustering and Robust Variance Estimation . . . . . . . . . . .. . . . . . . . 62
6.4 Estimating the Poisson Model . . . . . . . . . . . . . . . . . . . . . . .. . . 64
6.5 Estimating the Negative Binomial Model . . . . . . . . . . . . . .. . . . . . . 67
6.6 Zero Inflated Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 Summary, Potential Drawbacks and Open Questions 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A Formulae Derivation 90
A.1 Formulae from Section 2.1.3 . . . . . . . . . . . . . . . . . . . . . . . .. . . 90
A.2 Formulae from Section 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . .. . . 90
ii
A.3 Formulae from Section 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . .. . . 93
A.4 Formulae from Section 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . .. . . 93
A.5 Formulae from Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . .. . 95
A.6 Formulae from Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . .. . 97
A.7 Formulae from Section 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . .. . 97
A.8 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 99
A.9 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 99
A.10 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 102
A.11 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 103
A.12 Data Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.13 Maximal Acceptable Commuting Distance, Simulations .. . . . . . . . . . . . 105
A.14 Zero-inflated negative binomial estimation . . . . . . . . .. . . . . . . . . . . 107
A.15 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 109
A.16 Withdrawals from the Labor Market . . . . . . . . . . . . . . . . . .. . . . . 111
A.17 Alternative Specifications . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 113
A.18 Software Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
iii
Preface
This dissertation was written during my time as a doctorate student at the University of Regens-
burg and work on the project ”Flexibilitat der Lohnstruktur, Ungleichheit und Beschaftigung -
Eine vergleichende Mikrodatenuntersuchung fur die USA und Deutschland” (under supervision
of Prof. Dr. Moller), which is part of a greater project of the DFG ”Flexibilisierungspotenziale
bei heterogenen Arbeitsmarkten”.
I wish to thank Prof. Dr. Joachim Moller and Prof. Dr. Lutz Arnold for their supervision and
valuable comments and suggestions, which in my view helped to improve my thesis and added
much to its scientific value. I benefited greatly from the discussions at the econometrics semi-
nar and lunch seminar held at the University of Regensburg, especially with Dr. Jorg Lingens,
Prof. Dr. emer. Walter Oberhofer, Prof. Dr. Rolf Tschernig (all University of Regensburg), Dr.
Johannes Ludsteck (IAB), Dr. Thomas Schreck (BulwienGesa AG, Munchen-Berlin). I also
wish to thank Prof. Bernd Fitzenberger, PhD (University of Freiburg) for his thorough com-
ments on one of my papers, which contributed much to this dissertation. I also enjoyed valuable
discussions with Prof. Dr. Wolfgang Franz, Melanie Arntz, Alfred Garloff (all ZEW), Dr. Ralf
Wilke (University of Leicester), Prof. Michael Burda, PhD (Humboldt University, Berlin). I
also wish to thank my new colleagues at ZEW for their help: Dr.Stephan Lothar Thomsen (for
his LaTeX expertise), Markus Clauss, Christian Gobel (forhis tip on using Gnumeric).
Of course, this work would have never been possible without constant care and support of my
family: my parents and grandparents, my sister Aisulu. And last, but certainly not least, I am
indebted much to my dearest wife – Olga, not only for her help with the LaTeX formula editor,
but most importantly for her love and understanding during all these years.
iv
Chapter 1
Introduction
Information economics has already had aprofound effect on how we think abouteconomic policy, and are likely to havean even greater influence in the future.The world is, of course, morecomplicated than our simple - or evenour more complicated models - wouldsuggest.
J. Stiglitz, Nobel Prize lecture,December 8, 2001
Job search theory is a relatively young actor on the stage of economics and is an integral part
of a broader field of economics of information. The ideas about functioning of the markets
where information is costly to obtain were first enunciated by the Nobel Prize winner G. Stigler
in his pioneering work ”The Economics of Information” published in the Journal of Political
Economy in 1961. To explicate the very essence of what the economics of information and job
search in particular all about I refer to Stigler himself:
I propose on this occasion to address the same kinds of questions to an entirely
different market: the market for new ideas in economic science. Most economists
enter this market in new ideas, let me emphasize, in order to obtain ideas and meth-
ods for the applications they are making of economics to the thousand problems
with which they are occupied: these economists are not the suppliers of new ideas
but only demanders. Their problem is comparable to that of the automobile buyer:
to find a reliable vehicle. Indeed, they usually end up by buying a used, and there-
fore tested, idea. Those economists who seek to engage in research on the new
ideas of the science - to refute or confirm or develop or displace them - are in a
sense both buyers and sellers of new ideas. They seek to develop new ideas and
1
persuade the science to accept them, but they also are following clues and promises
and explorations in the current or preceding ideas of the science. It is very costly
to enter this market: it takes a good deal of time and thought to explore a new idea
far enough to discover its promise or its lack of promise. Thehistory of economics,
and I assume of every science, is strewn with costly errors: of ideas, so to speak,
that wouldn’t run far or carry many passengers. How have economists dealt with
this problem? That is my subject. (G. Stigler, Nobel Memorial Lecture, December
8, 1982)
The first application of the economics of information, in particular the notion that information is
costly to obtain and returns in the future are uncertain, to the labor market appeared in Stigler’s
1962 work. Since McCall (1970) seminal article the job search theory has become a standard
tool for analyzing the decision making process of an unemployed individual who is looking for
work.
An important contribution of the job search theory is an interpretation of unemployment on a
microeconomic level. As Mortensen puts it:
The theory of job search has developed as a complement to the older theoretical
framework. Many writers found that the classic labor supplymodel with its em-
phasis on unilateral and fully informed choice could not explain important features
of the typical individual’s experience in the labor market.The experience of unem-
ployment is an important example. Within the income-leisure choice framework,
unemployment simply has no interpretation as a consequenceof the assumptions
that jobs are instantaneously available at market clearingwage rates known to the
worker. (Mortensen (1986, p.850))
The original contribution of the search theory was the theoretic approach to the analysis of un-
employment spell durations. In a partial equilibrium model(wages set by firms are considered
exogenous), which stems from developments in the theory of sequential statistical decision the-
ory, a worker is looking for a job in a decentralized labor market. Information on vacancies
and the pay is imperfect and must be acquired before a worker becomes employed. Viewing
this process as costly and sequential enables us to analyze variation in the unemployment spells
that workers experience and in the wages received once employed. The length of time a worker
spends looking for a job and the subsequent wage received once employed are both random
variables with distributions which depend on the worker’s individual characteristics as well as
those of the environment through conditions the worker determines for acceptable employment.
Because the framework has implications for the distribution of observables, likelihood functions
for estimating the econometric model can be derived from thetheory.
2
Applications of the search theory are rather broad. The partial equilibrium models concentrating
on a decision-making process of an unemployed individual enable to make implications for
the duration of unemployment, effect of unemployment insurance, mini-mum wages and etc.
Another strand of literature looks at the problem from a different angle. Burdett and Mortensen
(1998) analyze wage setting decision of firms within a searchmodel framework. This seminal
work illustrates why identical workers may receive different wages in equilibrium.
Full equilibrium models incorporate both decision-makingof an unemployed individual looking
for job and firm looking to fill a vacancy. Pissarides (1979) isone such example. In his work he
derives a full market equilibrium which satisfies worker equilibrium condition and firm equilib-
rium condition. Albrecht and Axell (1984) derive an equilibrium unemployment rate which also
satisfies worker and firm equilibrium conditions. Beside that, the authors were first to endoge-
nously determine the wage offer distribution, which resulted from workers’ different valuation
of leisure.
Search models have been extended in different ways. An important contribution was made by
Burdett (1978) who introduced the notion of the on-the-job search. The pioneering paper of Bur-
dett (1978) and other works following afterwards helped explaining the job-to-job transitions
and wage growth with the same employer. Another extension was introduced by Jovanovic
(1979). Originally, it was assumed that the package of characteristics attributed to the job is
known. In Jovanovic (1979), the worker does not know for surethe earning streams associated
with the job or some other relevant characteristics. The worker must spend some time on that
job to acquire all the relevant information. In this framework, the quit happens after the worker
has learned all relevant information about the job and considers it ”not a good match”, i.e. the
job did not met his expectations. Further extension to the basic model was the uncertainty about
the offered wages. In a standard search model, the moments ofthe wage distribution are as-
sumed to be known. Burdett and Wishwanath (1984) relax this assumption. In their model,
workers have an expectation on the mean wage and dispersion.When they apply for a job they
obtain information on the wage paid on this job. Every time they acquire information about
the next job they update their expectation of the mean wage and dispersion in a Bayesian way.
These extensions will not be formally presented in my work here and an interested reader is
advised to follow the references.
Most of the search model frameworks imply time-invariant reservation wages. Economic reality
suggests, however, that they are not. As early as in 1967, Kasper provided empirical evidence of
declining reservation wages over the search span. Attemptshave been undertaken to explain this
phenomenon theoretically. Gronau (1971) claimed that constant reservation wage hypothesis
does not hold if the infinite life horizon assumption is relaxed. However, as suggested by
Mortensen (1986) this is rather an aging effect which cannotexplain relatively large rates of
3
decline in reservation wages for relatively young workers reported in several studies. Hence,
with the exception of elderly workers close to the retirement age, infinite time horizon is not
a stumbling point. Mortensen (1986) provides an elegant explanation of declining reservation
wages by imposing a credit market constraint. The general nonstationary job search model can
be found in van den Berg (1990) where nonstationarity of reservation wages may arise due to
time-dependence of any exogenous variable.
Applications of search theory give predictions about individuals’ reservation wages, unemploy-
ment durations and reemployment opportunities. However, withdrawals from the labor market
have received so far unfairly little attention, despite being an important indicator of labor market
performance (Frijters and van der Klaauw (2006) is a notableexception). The dropouts were
already mentioned in the search model of McCall (1970). However, the McCall (1970) model
is static. This can well suit the decision making process whether to enter the labor market or not
but fails to explain the exits from the labor market of the already participating workers. One of
the objectives which I pursue in my work is to fill this gap by incorporating withdrawals from
the labor market into a nonstationary job search model, and as I will show, withdrawals from
labor force are a logical outcome of the nonstationary job search. I present a nonstationary job
search model with the possibility of withdrawal from the labor market in Chapter 4.1.3. The
outcome of the model is that lower reservation wages besidesshorter unemployment duration
also lead to higher exit rate from unemployment into nonparticipation. This tradeoff can be very
important for unemployment insurance policy. As the simulations in Chapter 4.1.3 show higher
unemployment compensation could ultimately lead to higheremployment.
Another important aspect which motivated this dissertation is an asymmetric change of the wage
offer distribution. In a standard search model (see for example Mortensen (1986)) reservation
wages increase with the mean of the wage offer distribution and with the mean-preserving
spread of the wage distribution. However, changing the spread of the distribution by holding
the mean constant implies a symmetric ”stretching” or ”compressing” of the distribution in the
tails. But what happens if the spread parameters for the lefttail of the wage distribution and
for the right tail may vary separately, which is usually the case when one faces the data? The
complication arising here is that changing the spreads in the left and right tail unproportionately
will affect the mean. To alleviate the problem I propose to use the notion of the median and
the median-preserving spread. The model presented in Chapter 5 shows how predictions of
the job-search theory change if asymmetric change of the wage distribution are allowed. The
empirical results based on German regional data support theresults of my theoretical model.
This new insight into the job-search theory is very important for the regional empirical analysis
where regional mean wage or dispersion are often used as regressors.
The structure of the dissertation is as follows: in Chapter 2, I present an overview of exist-
4
ing job-search literature with detailed description of selected works, which in my view are most
innovative. In my dissertation, I concentrate on job searchof the unemployed individuals, there-
fore Chapter 2 does not contain models allowing search on thejob (for search on the job see
a seminal paper Burdett (1978), respective chapters in Mortensen (1986) and Manning (2003)
and citations therein. Chapter 4.1.3 presents a nonstationary search model with declining reser-
vation wages. In this chapter I show the existence of a tradeoff between unemployment and
participation, which is supported by simulations. Chapter4 gives an overview of the econo-
metric methods used to estimate duration models. Moreover,in this chapter the problem of
an unemployment-participation tradeoff is addressed and methods of correcting the estimated
failure rates are proposed. Chapter 5 gives a locational job-search model with a possibility
of commuting with asymmetric changes of the wage offer distribution. Chapter 6 gives an
overview of the estimation methods for count data and presents the results of the estimation of
the German commuting data. Chapter 7 concludes, points out to potential drawbacks and open
questions.
5
Chapter 2
Review of Literature
The original contribution of the search theory was the theoretic approach to the analysis of
unemployment spells durations and dispersion of incomes. Theory of job search uses the tools
of sequential statistical decision theory for the typical worker’s problem of finding a job in
a decentralized labor market. Information on vacancies andwages associated with them are
considered as imperfect. This information has to be acquired before a worker can become
employed. Viewing the process of acquiring information as costly is an important contribution
of the economics of information and search theory. Because the search is costly an unemployed
worker has to seek an optimal strategy which maximizes the present value of his future returns.
Hence, with search costs present and time discounting no rational worker has an incentive to
wait indefinitely for an opportunity to be employed. Besides, since the market is imperfect job
offers are not immediately available. This explains the variation in unemployment durations in
a search theoretical framework.
Since the pioneering work of Stigler (1962), search models have been widely used in labor
market theory. Pissarides (1979) develops a search model inthe presence of an employment
agency. The model is grounded on rather restrictive assumptions, there is no wage variability
and the separation rate is exogenous. The most important feature in his model is that two
alternative search methods are possible - random search andsearch via an employment agency.
The author shows that encouraging random search would increase the overall matching rate.
In Hall (1979) there are no intermediaries, i.e. job-searchers and employers approach each
other directly. Hall (1979) relaxes the assumption of a constant separation and job-finding
rate. He proposes the existence of an efficient separation rate and an efficient job-finding rate
as solutions to the maximization problem. Market equilibrium, given an efficient separation
and job-finding rate, gives the natural unemployment rate. The classical search model was
formulated by McCall (1970) where unemployed workers received job offers drawn from a
non-degenerate wage offer distribution. The reservation wage in this model is a solution to an
intertemporal maximization problem.
6
Another strand of search literature deals with search across spatial units. In Burda and Profit
(1996), for example, agents do not search for vacancies onlyin their area of residence but can
search in other districts as well. In this model distance is an important factor affecting the
search effort. Individuals are set to optimize their searchintensity taking the behavior of others
as given. There are several works on commuting which are based on spatial search theoretical
models (see e.g. van Ommeren and van der Straaten (2005)). Unlike in classical search model,
which assumes wage dispersion, van Ommeren and van der Straaten (2005) allow for constant
wages but dispersion of distances to work. Unemployed individuals in their model solve for
maximal acceptable travel distance.
Many search theorists sought for models that could explain the dispersion in wages. Albrecht
and Axell (1984) assumed two types of workers with each type having a different value attached
to leisure. Because of differences in the value of leisure, reservation wages differ. The authors
derive two reservation wages for the two types of workers. Firms are heterogeneous in Albrecht
and Axell (1984) and differ in their productivity. The density function of productivities is as-
sumed to be non-decreasing. Albrecht and Axell (1984) showsthat differences in the value of
leisure generate different wages offered by firms in equilibrium. Hence, the wage offer distribu-
tion is endogenously generated in the model. Eckstein and Wolpin (1990) extend the model of
Albrecht and Axell (1984) to allow for more than two types of workers and endogenous wage
offer probability which depends on the number of active firmsin the market. In this model,
although workers are homogenous in productivity, wage dispersion emerges due to differences
in workers’ tastes for leisure and efficiency of firms measured in terms of output per worker.
One of the most famous search models is probably that of Burdett and Mortensen (1998). The
authors show that in the presence of the on-the-job search the dispersion of wages could arise
even if all workers and firms are identical. Burdett and Mortensen (1998) and Albrecht and Ax-
ell (1984) models were synthesized by van den Berg and Ridder(1998) to account for workers
and firms heterogeneity as well as for the on-the-job search.They also impose a legal minimum
wage in their model. Similar approach is undertaken by Bontemps, Robin, and van den Berg
(2000), who derive admissible endogenous wage distributions for a large class of productivity
distributions (see also Bowlus and Grogan (2001)).
The literature review presented in this section uses notations of the original papers adapting it
when necessary for the sake of conformity.
7
2.1 Models with Constant Wages (Urn Models)
2.1.1 Full Equilibrium Model with Employment Agency
This section describes the model of Pissarides (1979). Its important feature is that an employ-
ment agency is present in the model, although it is not alwaysan intermediary between firms
and searchers. Job searchers may choose to register at an employment agency and thus receive
offers from the mediator. ”Random search” is also availableto them, which means directly vis-
iting firms and making inquiries about available vacancies.It is assumed that the unemployed
always register (at the employment agency) to be entitled for the unemployment compensation.
Nevertheless, some part of these jobless workers may choosethe ”random search” to increase
their employment chances. It is necessary, however, to introduce the cost of random search into
the model, because otherwise all unemployed workers would be involved in a random search.
The principle idea is that agents choose the optimal mixtureof search methods to maximize the
value of search.
In the same fashion, the firms in the model can also optimize their behavior with two alternatives
possible: advertising the vacancies through the employment agency or privately. As Pissarides
puts it: ”...firms may change the intensity of their search byswitching to methods with higher
job-matching probability (which in general be more expensive). It is this variable intensity of
search that plays the crucial role in the determination of aggregate behavior in this model.”
(Pissarides (1979), p.819)
For simplicity the author assumes that all firms offer the same wage to rule out the search on
the job and rejection of job offers. Some fraction of the unemployed,S 6 U also searches
randomly (besides being registered at the agency) andx of them succeed. For the sake of
simplicity it is assumed that each firm opens only one position. Total number of firms isV . R
firms register their vacancy with the employment agency,A = V −R firms advertise privately,
thusA vacancies are available to random searchers.
Jobs are destroyed for exogenous reasons at a rateδ. Total separations are given byδ(L − U),
whereL is the total labor force. Each period the employment agency arrangesy matches.
The agents do not know which firm would be visited by other workers. Therefore, the optimal
strategy would be to choose a firm at random. The probability that a vacancy will not be
searched by any of thoseS workers is given by(1 − 1/A)S. And the probability that a firm in
the setA will find a worker is given bya = 1− (1− 1/A)S. The total number of job matchings
each period isx+y, wherex is the number of job matches through random search andy through
employment agency.
8
The unemployment equilibrium condition implies equating the job inflows and outflows:
x(S, A) + y(U, R) = δ(L − U), (2.1)
where the left-hand side gives the total number of matchingsand the right-hand side – the total
number of separations.
Worker Equilibrium
The probability that a registered individual will receive an offer from the agency per period is
given byq = y/U , and the probability that a random searcher will receive a job offers is given
by p = x/S. In this model workers are assumed to be risk-neutral.
The models shares standards assumptions in the literature:registered unemployed receive the
benefitsb, random search costsc. Moreover, the agency charges workers for the job placement
h.
DenoteW to be the lifetime returns of the employed worker andΩU - the returns of the unem-
ployed worker who is not searching randomly, which yields:
ΩU = b +q
1 + rW +
1 − q
1 + rΩU − q · h
W = w +δ
1 + rΩU +
1 − δ
1 + rW.
(2.2)
If they unemployed are engaged in a random search their valuefunction can be written as:
ΩW = −c +p
1 + r(W − ΩU ). (2.3)
Substituting forW andΩU one obtains:
ΩW =p
r + δ + q(w − b + q · h) − c. (2.4)
Firm Equilibrium
Using previous notations,R firms register their vacancies with the employment agency, and A
firms advertise privately. Letπ denote the profit (since a firm may employ only one worker it
is also profit per worker), andρ be the cost of capital needed for one job position per period,
which is a sunk cost that borne irrespective of whether the position is filled or vacant. If firms
choose to advertise privately, they pay advertisement cost, α per vacancy. The profit from a
filled position is thusπ − w − ρ per period, whereas the vacant position has a cost ofρ per
9
period. DenoteΠ as returns from a filled position andΩA as returns from a vacancy, which is
not registered at the employment agency. Then:
ΩA = −α − ρ +a
1 + RΠ +
1 − a
1 + rΩA
Π = π − w − ρ +δ
1 + rΩA +
1 − δ
1 + rΠ.
(2.5)
Denote the returns from a registered vacancy byΩR∗ , which gives:
ΩR∗ = −ρ +
g
1 + rΠ +
1 − g
1 + rΩR
∗ − g · υ
Π = π − w − ρ +δ
1 + rΩR +
1 − δ
1 + rΠ
ΩR∗ = ΩR + C,
(2.6)
whereΩR is the value of a vacancy before registration,C is a registration cost of a vacancy,
andυ is the fee that the agency charges the firm for a successful match. ”Firms will choose the
job-matching method that yields the highest returns. In equilibrium, if both methods are used
by different firms, both methods must be equally attractive,soΩA = ΩR.” (Pissarides (1979, p.
823))
In equilibrium:
ΩA = 0
ΩR = 0.
(2.7)
Market Equilibrium
Market equilibrium is given by the simultaneous satisfaction of the four equilibrium conditions:
ΩW = 0
ΩA = 0
ΩR = 0
−x(S, A) − y(U, R)− δU + δL = 0.
(2.8)
The first condition implies that net marginal returns from random search are zero. The next
two mean that the returns from opening up more vacancies are zero. The last one implies that
the unemployment pool is constant over time, i.e. inflows into unemployment pool equal the
outflow to jobs.
10
2.1.2 Equilibrium Unemployment Rate and Employment Duration
This section describes the model of Hall (1979). A distinctive aspect of Hall (1979) model
is that he stresses the notion of the unemployment and employment duration as an important
factor which determines the equilibrium unemployment rate. Hall introduces the concept of an
”efficient duration” of employment. For an employer the efficient duration depends on the cost
of recruiting and training; and for the worker it depends on the cost of finding new jobs. In
tight markets where jobs are easy to find, workers prefer shorter jobs but this imposes higher
recruiting cost on employers so they favor longer jobs.
Equilibrium Unemployment Rate
Suppose that jobs and workers are perfectly homogeneous. A vacancy in this model is instantly
filled, but the unemployed must wait until a job offer arrives, which is a stochastic process. The
unemployed accept the first job offer encountered. If they receive more than one, they accept
one at random because jobs are homogeneous with respect to wage.
Employers makeV offers toS job-seekers. The probability that a particular worker receives a
particular job is1/S. The probability that an agent will receive no offers at all is:
1 − f = (1 − 1/S)V =[
(1 − 1/S)−S]−V/S
, (2.9)
wheref is the job-finding rate. IfS → ∞, (1 − 1/S)−S → e, which yields the solution for the
job-finding rate:
f = 1 − e−V/S . (2.10)
Defineρ as the number of vacancies needed to be opened to generate onejob on average, which
is ρ = V/(f · S). From Equation 2.10 it follows thatV/S = − ln(1 − f), hence:
ρ(f) = − ln(1 − f)/f. (2.11)
The equilibrium is achieved when the flows into an out of unemployment are equated. Given a
separation rate,s, this can be given as:
S = (1 − f)S + s · E, (2.12)
or
S/E = s/f, (2.13)
11
whereE is the number of employed workers. The unemployment rate canbe given then as:
u =(1 − f)S
E + (1 − f)S=
s
s + f/(1 − f). (2.14)
It is obvious from Equation 2.14 that the unemployment rate increases with the separation rate
and decreases with the job-finding rate, which is intuitively clear.
Optimal Employment Duration
From the firm side there should exist an optimal contract length. Firstly, because firms would
dislike very short employment spells due to fixed costs associated with recruiting and train-
ing. Secondly, very long contracts make firms inflexible in adjusting their employment level
in response to economic situation as firms would have to pay workers the compensation for
premature contract termination and layoff. Hence, firms would pay lower wages for very short
contracts and lower wages for very long contracts and higherwages for some intermediate
length contracts. This could be given by an isocost line plotted against the contract length,
which is concave and has a maximum.
For workers the tradeoff also exists. Very short jobs may be too costly because looking for a
new job involves certain expenses. On the other hand very long jobs reduces workers’ flexibility,
hence workers would require a compensation (in terms of higher wages) for very short contracts
and very long contracts to be indifferent among the job offers with various contract lengths. This
would give an indifference curve for a worker, which is likely to be convex.1 The intersection of
an isocost and indifference curve gives an efficient wage andefficient contract length. Treating
the expected duration of employment as a reciprocal of a separation rate, this intersection gives
the solution for an efficient separation rate.
Efficient Job Finding Rate
Let λ(s) be the probability that a job will be filled as a function of theseparation rate. The
expected cost associated with the job isw · λ(s). The unconditional probability that a job is
unfilled is s. However, for a separation to occur, the vacancy must be filled. This gives the
conditional probability that a job is unfilleds · λ(s). To keep the job filled a firm needs to hire
at a rates · λ(s), which requires a flow of offers ofρ(f)s · λ(s).
Suppose that each offer costs the firmµ · w, soµ is the fraction of the wage. Then the firm is
interested in minimizing its cost function given by:
C(w, s, f) = w · λ(s) · (r · s · ρ(f) + 1). (2.15)
12
Moreover, suppose that workers are interested in maximizing theeffectiveincome, where effec-
tive income is defined as:
y = (1 − u)w =f
(1 − f)s + fw. (2.16)
So the effective income is the expected wage conditional upon being employed;u can be also
defined as the fraction of time agents expect to be unemployed. The equation in 2.16 imposes a
constraint onto 2.15, hence the efficient job-finding rate can be given as:
f = arg min
(
1 + s1 − f
f
)
y · λ(s) · (µ · s · ρ(f) + 1)
, (2.17)
since bothy andλ(s) are independent off , Equation 2.17 simplifies to:
f = arg min
(
1 + s1 − f
f
)
· (µ · s · ρ(f) + 1)
. (2.18)
In the similar fashion, the efficient separation rate solvesfor:
s = arg min(f) = arg min
(
1 + s1 − f
f
)
λ(s) · (µ · s · ρ(f) + 1)
. (2.19)
The system of two equations 2.18 and 2.19 with two endogenousparameterss andf guarantee
the unique solution to the natural unemployment rate given in 2.14.
The model of Hall (1979) solves for the natural unemploymentrate as the outcome of efficient
employment arrangements. Hence, in this sense, the naturalunemployment rate is socially
optimal unemployment.
2.1.3 Spatial Search with Constant Wages
This section describes the model of Burda and Profit (1996). In their model, Burda and Profit
(1996) consider an individual who can determine his search activity in two dimensions: where
to search and how many jobs to apply for. Each spatial unit hasa job agency which mediates
contacts between searchers and potential employers. Workers apply randomly to firms. Once
applied, an individual is invited to an interview after which it is decided whether he is accepted
or not. There is a cost,c, associated with each interview. If he applies for a job in the regionj
different from the region of origin, say,i, then interview cost is increased toc + a ·Dij. Where,
Dij is the distance between the regionsi andj in kilometers anda is thus a per-kilometer cost of
travel. Individuals assume to optimize their search intensity in order to maximize their expected
net income. Letf denote the job finding probability,r - the interest rate, andm - the search
13
intensity, which can be thought of as number of jobs to apply for. Then, the objective function
for an individual searching in the regionj is given by:
maxmj
[1 − (1 − fj)mj ]w/r − mj(c + a · Dij). (2.20)
The first term in Equation 2.20 is the expected benefit from making mj interviews in the region
j and the second term is the total cost of those interviews. Assuming for simplicity that the
expected income in unemployment equals zero, the authors derive the solution:
mj =
f−1j · ln
(
fj(w/r)/(c + a · Dij))
for fj(w/r)/(c + a · Dij) > 1
0 otherwise.(2.21)
From Equation 2.21 it is seen that optimal search intensity is increasing in the wage and decreas-
ing in discount rate and the cost of applying for the job. The effect of job finding probability
on search intensity is ambiguous. One the one hand, it increases the expected benefits, but on
the other hand, at given relative returns, less search is necessary to achieve the same expected
benefits.
The model of Burda and Profit (1996) uses the same search principle as in Pissarides (1979).
The wage is constant, thus jobs do not differ from one another, and therefore application to
firms is a random process. However, in their model agents optimize their search intensity in
each spatial unit given the distance between the residence location and potential employment
location and difference in job-finding probability across spatial units.
2.2 Exogenous Wage Dispersion
2.2.1 Single Wage Offer Model, Discrete Time
This section describes the model of Franz (2006). The necessary condition for an unemployed
individual to accept a job offer is that the wage offer exceeds his reservation wage:w > wR.
Suppose that an individual receives a wage offer with probability q, which depends on the
general situation in the labor market and personal characteristics, such as age, sex, qualification
etc (which are represented by a vectorz). Moreover, it is assumed that for each individual the
chances of getting a job are decreasing with the wage offer, due to increasing competition for
high-paid jobs. The probability of a successful match is given as:
p(z, wR) =
∫
wR
q(z, w)f(w)dw. (2.22)
14
The expected wage is given as:
E(w|w > wR) =
∫
wR
wq(z, w)f(w)dw
∫
wR
q(z, w)f(w)dw. (2.23)
The expected wage is a conditional expectation, given that the wage offer exceeds the reserva-
tion wage. The present value of the reservation wage is:
∞∑
t=0
wR
(1 + r)t=
(1 + r)wR
r. (2.24)
Denote the unemployment benefits, which an agent receives staying unemployed byb and the
constant search cost per period asc, the present value of expected wages (if a person gets a job)
and costs of search equal:
(b − c)(1 + r)
r + p(z, wR)+ p(z, wR) · E(w|w > wR)
1 + r
r(r + p(z, wR)). (2.25)
In the optimum the present value of acceptable wage offers must be equal the present value
of the returns to search. Knowing that an agent accepts only those wage offers exceeding
his reservation wage, in the optimum the discounted returnsto search equal the discounted
reservation wage, i.e. Equation 2.24 equals Equation 2.25.Solving it forwR gives:
wR =r(b − c) + p(z, wR)E(w|w > wR)
r + p(z, wR). (2.26)
This relationship shows that the reservation wage decreases with search costs, increases with
unemployment benefits and wages. Also a lower probability ofreceiving and accepting a wage
offer, due to personal characteristics or labor market conditions, reduces the reservation wage.
It is easy to derive from Equation 2.26. In fact:
∂wR
∂p(z, wR)=
E(w|w > wR)(r + p(z, w)) − r(b − c) − p(z, wR)E(w|w > wR)
(r + p(z, wr))2=
r(E[
w|w > wR]
+ c − b)
(r + p(z, wR))2> 0,
(2.27)
as the value of employment must exceed the value of unemployment; otherwise agents would
not accept job offers at all and remain constantly unemployed.
15
2.2.2 Multiple Wage Offer Model, Continuous Time
This section is based on the model presented in Mortensen (1986), which is an extension of
McCall (1970). A distinct feature of this model is that besides wage variability it considers that
in case of multiple offers agents pick the offer with the highest wage. The model assumes that
workers live forever and there are no separations and quits.No recall is possible, i.e. once job
offer is declined, an agent cannot go back to the offer and accept it.
Consider an agent who looks for a job. Suppose the search costs him c per time period and
β(h) is the discounting factor for timeτ . The distribution of wages is given asF (w). In a given
period an agent could receiven job offers, withn being a random variable which follows the
Poisson distribution:q(m, τ) =e−λτ (λτ)m
m!, whereλ is the offer arrival rate, which is assumed
to be exogenous. Discounting is assumed to be continuous andthe discount rate is given as:
β(τ) = e−rτ . The workers are assumed to be wealth maximizers, which is equivalent to utility
maximizing in risk-neutral case. If an agent receives multiple offers he accepts the one with
the highest wage. Definingwm = maxw1, w2, ..., wm, the distribution of wages accepted
by an agent is an extreme value distributionG(wm). Assuming thatF (w) andq(m, τ) are time
invariant, the value of search becomes time independent andthe Bellman equation could written
as:
Ω = (b − c)τ + β(τ)
[
∑∞m=1 q(m, τ)
∞∫
0
max [Ω, W (w)] g(wm)dw + q(0, τ)Ω
]
=
= (b − c)τ + β(τ)
[
∑∞
m=1 q(m, τ)∞∫
0
max [0, W (w)− Ω] g(wm)dw + Ω
]
.(2.28)
whereΩ denotes the value of search,b - value of leisure per period,W (w) - value of employ-
ment. The value of employment can be written as:
W (w) = wτ + β(τ)W (w) =wτ
1 − β(τ). (2.29)
It could be seen thatW (w) is monotonically increasing in wage. The value of search as afunc-
tion of a wage satisfies Blackwell’s sufficient conditions for a contraction: (1)Ω is monotonic in
w, (2)Ω is discountable. This implies that only one fixed point exists whereΩ(W (w)) = W (w).
Defining the reservation wage as the wage at which an agent is indifferent between accepting a
job and continuing search, this implies:W (wR) = Ω, which is unique, wherewR stands for the
reservation wage.
16
The probability of receiving more than one offer within an infinitesimal interval of time is
virtually zero (τ → 0). Hence, in continuous time, the Bellman equation in 2.28 considerably
simplifies:
rΩ = (b − c) + λ
∞∫
0
max[0, W (w) − Ω]dF (w). (2.30)
Knowing that:w = rW (w) andW (wR) = Ω, we obtain:wR = rΩ, which yields:
wR = b − c +λ
r
∞∫
wR
(w − wR)dF (w). (2.31)
Applying integration by parts to Equation 2.31 yields:
wR =r
r + λ(b − c) +
λ
r + λE(w) +
wR∫
0
F (w)dw. (2.32)
One could see from Equation 2.32 that the reservation wage depends on the wage distribu-
tion F (w). Usually the distribution is characterized by the first two moments: the mean as a
location parameter and the variance as a scale parameter. Inthe search literature it is conven-
tional to use the mean-preserving spread as a scale parameter which characterizes the shape
of the distribution due to Rotschild and Stiglitz (1970). Let s1 and s2 be mean-preserving
spreads of the wage offer distributionF (w). By definition of the mean-preserving spread,∫ ∞
0wdF (w; s1) =
∫ ∞
0wdF (w; s2). It can be shown that
∫ x
0F (w; s1)dw >
∫ x
0F (w; s2) for
s1 > s2, which implies that the reservation wage is:
• increasing in the value of time, i.e. if leisure becomes morevaluable, agents increase their
reservation wages;
• decreasing in the cost of search, i.e. when search becomes more costly, agents are willing
to accept lower-paid jobs;
• increasing in the mean of the wage offer distribution; the magnitude of the effect is less
than unity in absolute value, i.e. when the mean of the wage offer distribution goes up the
reservation wage will also increase but not as much;
17
• increasing in the mean-preserving spread, i.e. when dispersion of wages go up, agents
would not accept lower-paid jobs anymore; this is ”. . . the consequence of the fact that the
worker has the option of waiting for an offer in the upper tailof the wage distribution”
(Mortensen (1986, p. 865)).
2.3 Endogenous Wage Dispersion
2.3.1 Wage Dispersion Due to Worker Heterogeneity
This section is based on the paper of Albrecht and Axell (1984). An important feature of their
work is that unlike many previous models, in their setting the wage distribution is endogenously
determined. Moreover, they introduce heterogeneity of workers who differ in their value of
leisure.
Individuals
Individuals are assumed to maximize their lifetime utility. Let the utility function be given in
the form:
U = C + νL, (2.33)
whereU stands for utility,C for consumption,L denotes leisure, and parameterν is attributed
to a ”consumption value” of leisure.L in the model is a binary choice variable taking the value
of zero when an agent is working and one when he is searching. Individual’s wage rate is
denoted byw; and a non-wage income consists of ”dividends”,θ, which do not depend on the
individual’s employment status;2 unemployment compensation is denoted byb. Individual’s
utility choice becomes:
U =
w + θ if working at a wagew
θ + b + ν if searching.(2.34)
The economy in the model in any period consists ofk individuals andn firms. Firms are
assumed to live forever, but individuals exits the economy at the end of any period at a rateτ .
The key assumption of the model is that there are two types of individuals in the economy who
differ in the value they attach to leisure.
18
Consider a two-point wage distribution in the model withw0 andw1 denoting the low and the
high wage respectively, and letγ be the share of firms offering the low wage. In equilibrium,
w0 must be the reservation wage of those individuals who derivelow utility from leisure, and
w1 must be the reservation wage of those who derive high utilityfrom leisure.
Individuals who impute low value to leisure will accept the first wage offer encountered, whereas
those who impute high value to leisure will search until theyare offeredw1. Let β denote the
fraction of individuals with low value of leisure. The amount of search in the economy - mea-
sured by the unemployment rate - is therefore an increasing function ofγ.
Consider a search behavior of an individual with the value ofleisureν0 who has drawn a wage
w0. If he rejectsw0, then he ”earns” leisure worthν0, a non-wage incomeθ + b, and with
probability 1 − τ draws a wage in the next period. The wage sampled from a subsequent
drawing equalsw1 with the probability1 − γ; and if w1 is actually drawn, then it is accepted,
resulting in an expected future life-time utility of(w1 + θ)/τ . Otherwise, he continues the
search.
The value of rejectingw0 is therefore:
V = ν0 + b + θ + (1 − τ)
[
(1 − γ)(w1 + θ)
τ+ γV
]
=
=ν0 + b
1 − γ(1 − τ)+
(1 − γ)(1 − τ)
1 − γ(1 − τ)
w1
τ+
θ
τ.
(2.35)
Naturally,w0 is the reservation wage for the individual with the value of leisureν0. Hence:
V = (w0 + θ)/τ. (2.36)
Solving Equations 2.35 and 2.36 forw0 yields:
w0 =(1 − γ)(1 − τ)
1 − γ(1 − τ)w1 +
τ(ν0 + b)
1 − γ(1 − τ). (2.37)
If w1 is the reservation wage forν1 individuals, then in equilibrium the value of rejectingw1
must equal the value of acceptingw1:
(ν1 + b + θ)/τ = (w1 + θ)/τ ⇒ w1 = ν1 + b. (2.38)
19
Equations 2.37 and 2.38 give the solution for the wage distribution. In fact, one could see that
wages are endogenously determined by the set of exogenous parameters.
Firms
Firms produce according to the linear production functiony = p ·ℓ, whereℓ denotes the amount
of labor a firm hires andp is the productivity of labor, which is distributed across firms according
to a distribution functionA(p), with the corresponding density functiona(p). p is normalized
to lie within a unit interval. Letℓ(w) be per period labor supply to a firm offering a wagew,
then the profit of this firm solves for:
Π(w, p) = (p − w) · ℓ(w). (2.39)
For a firm to be profitablep > w must hold. So firms whose productivity is less than the
lowest reservation wage,w0, are out of the market, therefore only a fraction1−A(w0) is active.
Among active firms a fractionγ offersw0 and a fraction1 − γ offersw1. A profit-maximizing
firm needs to choose between offering a wagew0 or w1.
The authors further define a wage-indifference point,p∗. A firm with productivityp∗ is indiffer-
ent between offeringw0 or w1. Consequently, firms withw0 < p 6 p∗ will offer w0 and firms
with p∗ < p < 1 will offer w1. The equilibrium condition is thus:
γ =A(p∗) − A(w0)
1 − A(w0). (2.40)
The nominator reflects the share of firms offeringw0 amongall firms. Dividing it by 1 −A(w0) gives the share of firms offeringw0 amongactivefirms. The solution for ”indifference
productivity” is:
p∗ =w1ℓ(w1) − w0ℓ(w0)
ℓ(w1) − ℓ(w0). (2.41)
Next, Albrecht and Axell (1984) deriveℓ(w0) andℓ(w1). If a firm offersw0, only individuals
with the low value of leisure will accept that. Each periodτ · k · β individuals with low value
of leisure enter the economy. Lettingµ ≡ k
n
[
1 − A(w0)]
, there areτ · µ · β individuals with
20
the value of leisureν0 per active firm entering the economy each period. All of thosesearchers
contacting a firm will accept an offer. Hence,ℓ(w0) can be computed as the sum of theτ · µ · βagents who accept the low wage in the current period, the(1− τ)τ · µ · β surviving individuals
who acceptedw0 in the previous period, and so on. Therefore:
ℓ(w0) = τµβ[
1 + (1 − τ) + (1 − τ)2 + ...]
= µβ. (2.42)
For a firm offeringw1, all individuals contacting this firm accept the offer and the number of
contacts per firm per period is the sum of:
τµβ ν0 individuals entering the economy
τµ(1 − β) ν1 individuals entering the economy
τµ(1 − β)γ(1 − τ) ν1 individuals who have searched once
τµ(1 − β)γ2(1 − τ)2 ν1 individuals who have searched twice,
and so on. . .
The resulting sum is then:
τµβ + τµ(1 − β)[1 + γ(1 − τ) + γ2(1 − τ)2 + ...] = τµβ +τµ(1 − β)
1 − γ(1 − τ). (2.43)
As a resultℓ(w1) = µβ +µ(1 − β)
1 − γ(1 − τ).
The derivation of the equilibrium unemployment rate is rather straightforward. In any period
there areτ · k(1 − β)γ agents who search for the first time,τ · k(1 − β)γ2(1 − τ) agents who
search for the second time, and so on. Thereby, the equilibrium unemployment rate can be given
as:
u = τ(1 − β)γ[1 + γ(1 − τ) + γ2(1 − τ)2 + ...] =τ(1 − β)γ
1 − γ(1 − τ), (2.44)
with
21
du
dγ=
τ(1 − β)
[1 − γ(1 − τ)]2> 0. (2.45)
So the equilibrium unemployment rate is an increasing function of the share of firms offering a
low wage as required. The cutoff productivity can then be derived as:
p∗ =w1ℓ(w1) − w0ℓ(w0)
ℓ(w1) − ℓ(w0)= w1 +
(w1 − w0)ℓ(w0)
ℓ(w1) − ℓ(w0)= w1 +
(w1 − w0)β
(1 − β)/[1 − γ(1 − τ)]. (2.46)
Using Equations 2.37 and 2.38 one could establish thatw1 − w0 =τ(ν1 − ν0)
1 − γ(1 − τ), hence:
p∗ = v1 + b +τ(ν1 − ν0)β
1 − β(2.47)
The comparative statics show that the equilibrium unemployment rate increases with the un-
employment benefit. However, strikingly, if one can discriminate the unemployment insurance
between individuals with low value of leisure and high valueof leisure, increase in unem-
ployment compensation for agents who impute low value to leisure decreases the equilibrium
unemployment rate!
Efficiency Issues
Albrecht and Axell (1984) established that an increase in unemployment compensation leads to
higher unemployment for a broad class of productivity distribution functions. However, they
show further that the socially optimal level of unemployment is not necessarily zero.
Define the social objective function as per capita utility:
U∗ = C∗ + ν1u. (2.48)
The first term in Equation 2.48 represents per capita consumption and the second term is the
value of leisure per capita. Per capita production (which isequal to per capita consumption)
is the sum of production from low-wage firms and high-wage firms. Production from low-
wage firms is the product of (i) the number of firms offeringw0, which is equal ton[A(p∗)],
(ii) labor supply to low-wage firms,ℓ(w0) and (iii) the average productivity of firms offer-
ing w0, which is∫ p∗
w0
pdA(p)/[A(p∗) − A(w0)]. Hence the total production from low-wage
22
firms is nℓ(w0)∫ p∗
w0
pdA(p). In the same fashion total production from high-wage firms is
nℓ(w1)∫ 1
p∗pdA(p).
Equilibrium per capital consumption can be given thereforeas:
C∗ =n
k
ℓ(w0)
p∗∫
w0
pdA(p) + ℓ(w1)
1∫
p∗
pdA(p)
=
=1
µ[1 − A(w0)]
ℓ(w0)
1∫
w0
pdA(p) + [ℓ(w1) − ℓ(w0)]
1∫
p∗
pdA(p)
.
(2.49)
Or alternatively,
C∗ = β
p∗∫
w0
pdA(p)
1 − A(w0)+
(1 − β)(1 − γ)
1 − γ(1 − τ)
1∫
p∗
pdA(p)
1 − A(p∗). (2.50)
Defining the unemployment rate among theν1 individuals asu1 = τγ/[1 − γ(1 − τ)] yields:
C∗ = β
1∫
w0
pdA(p)
1 − A(w0)+ (1 − β)(1 − u1)
1∫
p∗
pdA(p)
1 − A(p∗). (2.51)
Then the per capita utility is given by:
U∗ = β
1∫
w0
pdA(p)
1 − A(w0)+ (1 − β)
1∫
p∗
pdA(p)
1 − A(p∗)− u
1∫
p∗
(p − ν1)dA(p)
1 − A(p∗). (2.52)
As stated before unemployment is likely to rise with unemployment compensation. It is clear
from Equation 2.52 that increase in the equilibrium unemployment rate leads to a utility loss.
However, one must keep in mind that the wage distribution also changes withb. Increase in
unemployment compensation raises the reservation wages. Searchers sort out less productive
firms which cannot match their wage aspirations and workers are consequently employed by
more productive firms. Hence, the change in the wage distribution drives inefficient firms out
of the market. This results in a utility gain. The total effect is unclear, but what is important
from this analysis is that unemployment compensation is notnecessarily socially undesirable.
23
2.3.2 Equilibrium Wage Dispersion with Identical Workers
The Burdett-Mortensen model is one of the major contributions to the theory of labor economics
of the last decade. The model was able to explain the long-standing question why observation-
ally equivalent workers are still paid different wages in equilibrium. As D. Margolis puts it:
Like researchers in many other fields of science, some labor economists have been
looking for a ”universal theory of everything”, or at least insofar as concerns labor
market outcomes like employment and unemployment, wage distributions, firm
size, seniority returns, and so on. The enthusiasm with which the literature has
adopted the Burdett-Mortensen (1998) and Pissarides (2000) frameworks suggests
that some macro-based labor economists, especially in Europe, believe they have
found their holy grail. (D. Margolis, Annotation to the bookof Mortensen (2003)).
This section is based on a seminal paper of Burdett and Mortensen (1998) (a nice overview
of the Burdett and Mortensen (1998) model can be found in Manning (2003) from where the
notation is borrowed). Consider an economy withMF firms andMW workers. LetMF and
MW be fixed and denoteM ≡ MF /MW . It is assumed that each firm opens but one vacancy.
The inflow of job offers to unemployed workers is a stationaryPoisson process with the arrival
rateλ. Employed workers may search on the job. Workers are homogeneous. Let the arrival
rate of job offers to employed workers be alsoλ. Jobs are also destroyed for exogenous reasons
at rateδ. Equilibrium unemployment rate is thenu =δ
δ + λ.
The fraction of workers receiving wagew or less is given by:
G(w; F ) =δF (w)
δ + λ(1 − F (w)), (2.53)
whereF (w) is the distribution of wage offers, which is to be endogenously determined. The
separation rate of workers in a firm paying a wagew is δ+λ(1−F (w)). The first term represents
the flow of employed workers into non-employment and the second term is the flow of workers
to other firms offering a higher wage thanw.
The inflow of workers to a firm paying a wagew is:
λ
M[u + (1 − u)G(w; F )] =
δλ
M [δ + λ(1 − F (w))], (2.54)
whereλu/M is the rate of recruiting from non-employment andλ
M(1 − u)G(w; F ) is the flow
of workers from other firms paying less thanw.
The steady-state labor supply of workers to a firm is then:
24
L(w; F ) =δλ
M [δ + λ(1 − F (w))]2. (2.55)
Employed workers are identical in productivity. Hence, anyworker employed at a firm gener-
ates a product worthp. The profits of a firm solve:
π(w; F ) =δλ(p − w)
M [δ + λ(1 − F (w))]2. (2.56)
Each firm sets a wage to maximize profits. In steady-state profits of firms should be equal irre-
spective of the wage set to preserve a non-degenerate wage distribution.3 Suppose that workers
value their leisure atb, which would be the reservation wage for non-employed workers. Then
the steady-state level of profits is given by:
π(w; F ) =δλ(p − b)
M [δ + λ]2. (2.57)
This gives solution to the equilibrium wage offer distribution:
F (w) =δ + λ
λ
[
1 −√
p − b
p − w
]
, (2.58)
and the distribution of observed wages:
G(w) =δ
λ
[√
p − b
p − w− 1
]
. (2.59)
The expected observed wage in the economy is then:
E(w) =δ
δ + λb +
λ
δ + λp. (2.60)
The lowest observed wage in the economy is henceb and the highest isp − (p − b)( δ
δ + λ
)2
.
25
The phenomenon of wage dispersion has been attracting much attention of many labor economists.
Human capital theory attributed these differences in pay tovariation in human capital across
workers (see the seminal work by Mincer (1974)). However, ”.. . observable worker character-
istics that are supposed to account for productivity differences typically explain no more than 30
percent of the variation in compensation across workers. . .” (Mortensen (2003, p. 1)). Although
controlling for firm-specific effects can explain about 70% of wage variation the question still
remains: Why firms follow different wage policies and why similar workers within one firm
are still paid differently? The seminal paper by Burdett andMortensen (1998) presents a model
where identical workers are still paid differently in equilibrium. The model has been modified
in several ways to make predictions about the shape of the wage distribution more consistent
with observed data (see e.g. Bontemps, Robin, and van den Berg (2000) and van den Berg
and Ridder (1998)); but even in its simplest form the model still sheds much light onto the
phenomenon of wage dispersion.
2.4 Conclusion and Empirical Relevance
Models presented in this chapter give an overview of developments in the job search theory.
Search theory has been successfully applied for various empirical questions. For example,
Burda and Profit (1996) apply the theory of locational searchto estimate the matching functions
using the Czech data (the rate at which unemployed workers are matched with the available
vacancies). Kiefer and Neumann (1979) empirically test thehypothesis of the search theory
that the reservation wage is constant over time.
Most interesting are probably the so-called structural models. For example, in the unem-
ployment duration analysis, each parameter of the so-called hazard function (see Section 4.1)
λ(1 − F (wR)) is estimated. Using the functional relationship between unemployment dura-
tion and the hazard function, the reservation wage and otherexogenous variables, and wage
offer function and observed wages it is possible to estimateseparatelyλ, wR, andF (w), which
are called structural parameters (therefore the notion ”structural model”). For identification of
the model reservation wages should be observed4 (or partially observed) and structural form
restrictions are applied. Therefore not many datasets allow identification of the structural pa-
rameters of the model. A recent example of the structural model can be found in Frijters and
van der Klaauw (2006). Authors introduce the nonparticipation option into van den Berg (1990)
model (a special case of this model can be found in Section 3.1). The authors use the GSOEP
data for the period 1989-1995. The main difference of this empirical work is that the authors
explicitly allow for exits into nonparticipation. Frijters and van der Klaauw (2006) define the
withdrawals as being coded in the GSOEP data by ”maternity leave”, ”housewife or house-
26
husband” or ”other”. The reservation wages of the unemployed individuals are reported in the
data (if an individual was unemployed at the date of the interview). Unemployment benefits
are partially observed and are comprised of unemployment insurance and unemployment assis-
tance. If not, the authors impute them by regressing the log of benefit on the previous wage
and other individual characteristics. Using the GSOEP datathe authors aim at identifying the
wage offer distribution, the job offer arrival rate, the discount rate, and the instantaneous utility
of nonparticipation. The wage offer distribution above thereservation wage can be identified
through the accepted post-unemployment wages after the unemployment spell. The tail below
the reservation wage cannot be identified (see Flinn and Heckman (1982)). Knowing the dis-
tribution of offered wages and the reemployment hazard one could identify the arrival rate (up
to certain normalization). The instantaneous utility of nonparticipation can be identified from
the length of the unemployment spell until withdrawal from the labor force and the reservation
wage before withdrawal. The discount rate can be identified if the reservation wage and its first
derivative is observed. The authors find that the wage distribution facing unemployed workers
shifts downwards with the unemployment spell. The authors argue that the shift of the wage dis-
tribution can be attributed to the loss of skills. Moreover,they find that the fastest loss of skills
occurs during the first year of unemployment. Hence, the authors suggest that the measures
aimed at halting the loss of skills should be taken during thefirst year of unemployment.
An important phenomenon, which is yet missing in the nonstationary job search models, in my
view, is a tradeoff existing between duration of unemployment and withdrawals from the labor
market. In the next chapter I will demonstrate the existenceof a tradeoff between unemployment
and participation. Hence, a change in a variable which results in reduction in unemployment
duration would result in more exits into nonparticipation and therefore reduce the participation
rate. This tradeoff poses certain problems for comparison of labor market performance and esti-
mation of survival rates. Empirical methods for correctingthe estimated job-finding probability
will be discussed in Section 4.2. The relevance of this tradeoff to policy issues is also discussed.
27
Chapter 3
Nonstationarity in the Theory of
Job Search and Withdrawals from
the Labor Market
3.1 Theoretical Framework
The model built here is in principle a dynamic variant of McCall (1970) (see Section 2.2.2).
Unemployed workers are identical and live forever. They possess the knowledge about the
parameters of the wage offer distribution, but they have no information when job offers arrive
and what wages are associated with them. Once accepted by a firm, workers must immediately
reply (accept the job or decline), so no waiting is allowed. Once the job is rejected it cannot be
recalled. Searching involves a direct costc per period. Hence, by not participating agents can
always enjoy the ”pure” leisure (without incurring the costc). The arrival rate is assumed to be
declining over time, which is the special case of van den Berg(1990). The probability that an
agent receives job offers is given by a Poisson probability distribution:
q(m, τ, λ(t)) =e−λ(t)τ (λ(t)τ)m
m!. (3.1)
Searchers discount at a rateβ(τ). When an agent receivesm job offers, he picks the best one.
Define wm = maxw1, w2, ...wm. The distribution of accepted offers is an extreme value
distributionG(wm) with a density functiong(wm).
The Bellman equation for the optimal value of search can be given as:
28
Ω(t) − (b − c)τ =
β(τ)[
∞∑
m=1
q(m, τ, λ(t))
∫ ∞
0
max [Ω(t), W (w)] dG(wm) + q(0, τ, λ(t))Ω(t + τ)]
=
β(τ)[
∞∑
m=1
q(m, τ, λ(t))
∫ ∞
0
max [0, W (w) − Ω(t)] dG(wm) +
q(0, τ, λ(t))Ω(t + τ) +
∞∑
m=1
q(m, τ, λ(t))Ω(t)]
. (3.2)
In continuous time the function of reservation wage can be given as:
wR(t) = b − c +λ(t)
r
∫ ∞
wR(t)
(w − wR(t))dF (w) +dwR(t)
r · dt, (3.3)
wherer stands for the discount rate. It is assumed that the arrival rate declines over time and
equals zero at timeT , formally, λ(t > T ) = 0 (in principle T could be equal to infinity).
If nonparticipation option is unavailable, the lowest possible reservation wage isb − c (the
unemployed has the option of rejecting all offers and keep searching). Consider now the case
when the unemployed may withdraw from the labor market, i.e.quit searching. In this case,
the unemployed will still ”earn” the value of leisure,b, (which is the utility of nonparticipation)
but without incurring the search costc. This implies that there is a critical time, denote it ast∗
, such thatwR(t∗) = b andwR(t > t∗) < wR(t∗) < wR(t < t∗). SincewR(t > t∗) < b it
is not optimal anymore for a worker to search aftert∗ and at that time the unemployed worker
drops out of the labor market. Thepotentialsearch time is then the maximum amount of time
an agent is willing to allocate to his search and is equal tot∗ .
Figure 3.1 shows the reservation wage as a function declining over time. Time at which the
reservation wage equals the value of leisure,b, is the potential search time. Att∗ the worker
drops out of the labor market (does not actively search) and receivesb. The reservation wage
equation can be rewritten as a differential equation:
wR(t) = max[
b, b − c +λ(t)
r
∫ ∞
wR(t)
(w − wR(t))dF (w) +dwR(t)
r · dt
]
. (3.4)
The model shows that att∗, when an agent drops out of the labor force, his reservation wage
equalsb. Solving the differential equation in 3.3 forwR(t) and substitutingb for wR(t∗) gives
the solution fort∗. The potential search time implies the maximum time an unemployed worker
is willing to allocate to his search. This means that aftert∗ the worker withdraws from the labor
29
t*
b−c
b
wR
5
3
1
t
6
4
2
0
543210
Figure 3.1: Declining Reservation Wage
market under condition that he has not accepted a job within this period. An important result
is that the dropout time is not a random variable, it is thechoice5 variable for searchers. The
potential search time,t∗, is known at any point in time and it does not depend on whetheran
unemployed worker finds a job or not, but isobservedonly if a worker fails to find employ-
ment until timet∗. Looking at Figure 3.1 one can see that exogenous factors which push the
reservation wage up also increase the potential search time.
3.2 Unemployment Participation Tradeoff
The instantaneous probability or hazard of transition fromunemployment into employment can
be written as:
φ(t) = λ(t)[
1 − F (wR(t))]
(3.5)
Measures which reduce reservation wages of the unemployed workers increase the hazard of
exit from unemployment into employment. However, as it was already mentioned in Section
3.1, lower reservation wage imply shorter potential search. Consider a group of identical work-
ers (call them group 1) with a search costc1. These workers have identical reservation wages
wR1 , and the length of potential search for them ist∗1. The fraction of exits from unemployment
30
into employment is1−exp(
−∫ t∗
1
0φ(t)dt
)
. Consider a second group of identical workers of the
same size for whom the search cost isc2, such thatc1 < c2, so thatwR1 > wR
2 andt∗1 > t∗2. The
second group has a higher hazard rate (of exiting from unemployment into employment) for a
given point in time aswR1 > wR
2 . However, this does not necessarily mean that theabsolute
numberof exits from unemployment into employment of the second group is larger. Workers
with reservation wageswR2 who have not accepted any job untilt∗2 withdraw from the labor
market. Workers with reservation wageswR1 do not drop out att∗2 but continue their search until
t∗1. Whether the number of transitions into employment duringt∗1 − t∗2 may compensate the
difference in the number of exits from unemployment into employment duringt∗2 is ambiguous.
Figures 3.2 and 3.3 show possible scenarios when reservation wages change.
Figure 3.2 shows the first possible scenario. The reservation wage of the first group iswR1
(dashed curve) and the potential search timet∗1, and the share of exits into employment isP1.
The reservation wages of the second group iswR2 (solid curve), the potential search time ist∗2
but the share of exits into employment isP2, which is greater thanP1. Figure 3.3 shows the
second possible scenario. Here the potential search of the second group ist∗2, such that the share
of exits into employmentP2 is smaller thanP1.
t2* t1*
P1
P2
10.0
0.8
0.6
7.52.5
prob
abili
ty
0.2
0.0
5.00.0
0.4
t
Figure 3.2: The tradeoff between the job-finding hazard and potential search time: Scenario 1
This tradeoff between the probability of finding a job and thelength of potential search poses
certain problems when one tries to compare performances of different labor markets. If we look
at Scenario 2 and consider only unemployment durations thenwe might get an impression that
31
t1*t2*
P1P2
10.0
0.8
0.6
7.52.5
prob
abili
ty
0.2
0.05.00.0
0.4
t
Figure 3.3: The tradeoff between the job-finding hazard and potential search time: Scenario 2
group 2 have better job-finding chances as the reservation wage of the group 2 is always above
that of the group 1 for any fixedt. However, once we incorporate the change in potential search
time, we could see that the overall job-finding probability of group 2 is lower.
3.3 Simulations
For the numerical simulations the function of the arrival rate was chosen to be linear in time and
be of the form:
λ(t) =
λ − t · k if t · k < λ
0 otherwise
Consider a pointT , such thatT · k = λ andλ(T ) = 0. The parameter values were chosen to
be:λ = 0.018, k = 0.01, T = 18 months. Moreover, the reservation wage function would have
the following form:
wR(t) = max[
b, b + α − c +λ(t)
r
∫ ∞
wR(t)
(w − wR(t))dF (w) +dwR(t)
r · dt
]
, (3.6)
whereα is the unemployment benefit which the unemployed worker receives while searching
but not in the case of nonparticipation. The wage offers wereset to be uniformly distributed for
32
the sake of simplicity with the infimumwL = 400 and supremumwU = 5000. Utility of non-
participation and search costs are chosen to be equal to 400 each. The amount of unemployment
insurance benefits (UI) was chosen to be equal 300 in the base model, in Model 1, UI = 350
and in Model 2, UI = 200. Reservation wage paths of three models are shown on Figure 3.4. In
Model 3, all parameters were chosen to be the same as in the base model except for the arrival
rate which is equal to 0.02 in Model 3. Theoretical considerations in Section 3.1 would imply
thatwR(T ) = b − c which would serve as an initial condition for solving the differential equa-
tion 3.6. For each model the potential search time and the cumulative job finding probability at
the time of withdrawal were calculated, which can bee seen inthe table below.
Table 3.1: Parameters and results of the simulated models
Model Unemployment Arrival Potential Cumulative
insurance (UI) rate (λ) search time job-find. prob.
Base 300 0.018 7.10 0.64
1 350 0.018 10.70 0.74
2 200 0.018 1.50 0.24
3 300 0.020 9.10 0.76
0
100
200
300
400
500
600
0 5 10 15 20
wR(t),M2 wR(t),M1 wR(t),base
Figure 3.4: Reservation wage paths
It is apparent from Figure 3.1 that an increase in the unemployment insurance benefits pushes
reservation wages up. However, Table 3.1 shows that the longer potential search induced by
an increase in the unemployment insurance benefits more thancompensates the negative effect
33
of higher reservation wages on employment prospects. Theseresults are partially in accord
with the empirical results of Frijters and van der Klaauw (2006). These authors find that a
reduction of unemployment benefits by 50% increases the percentage of individuals finding
a job within 12 months from 60.3 to 66.5% which is statistically insignificant. However, the
estimated effect of the benefits on the probability of an exitinto nonparticipation is positive (a
50% reduction of unemployment benefits reduced the share of exits into nonparticipation from
6.1 to 5.2%). This seems to be in discord with the theory. The problem could have arisen from
the definition of the unemployment insurance benefits. The definition of benefits in Frijters
and van der Klaauw (2006) covered the unemployment insurance benefits and unemployment
assistance (which is of unlimited duration). Recalling Equation 3.6 it seems more plausible that
unemployment insurance benefits affect onlyα and unemployment assistance affects the utility
of nonparticipation.
Concerning policy implications, the simulation results show that reducing the level of unem-
ployment insurance benefits may even worsen the situation due to a higher transition rate into
nonparticipation. Further work is required to identify theset of parameters which make unem-
ployment drop with higher unemployment compensation.
34
Chapter 4
Empirical Estimation of Duration
Models
4.1 Estimation Methods
Due to a tradeoff between unemployment and participation, as it was shown in Chapter 4.1.3,
unemployment duration or survival rate might not always be good indicators of labor market
performance. The empirical model presented in this chaptershows that care should be taken
when interpreting the results of a duration model estimation as the conditional job-finding prob-
ability estimated in a duration model may be significantly different from an unconditional job-
finding probability when the dropout rate is non-negligible.
The theoretical model set up in this work gives insight into the search process of workers. The
exogenous factors determine the reservation wage and thus the probability of finding a job.
Moreover, it has been shown that the parameters of the model affect the time an unemployed
spends looking for a job. Unfortunately however, there is noclosed-form solution for the reser-
vation wage, as it involves solving integrals with variablelimits of integration. Therefore, we
cannot determine the elasticities of reservation wage withrespect to the set of exogenous pa-
rameters analytically.
An offer is accepted only when the offered wage exceeds the reservation wage. Hence, the
probability that a worker is ”matched” at timet is given by:
∞∑
m=0
(λ(t)t)me−λ(t)t
m!
(
1 − F m(wR))
(4.1)
35
One could see that when no application is successful, i.e.m = 0, the probability in Equation 4.1
is zero. Equation 4.1 states that the probability of a match goes up with the arrival rate and goes
down with the reservation wage. Hence, without knowing the respective elasticities, the sign of
the effect of any parameter on the probability of a match cannot be determined. For instance,
when the arrival rate in the economy rises, agents will increase their reservation wages at the
same time, so they become pickier and could search longer until they have found a suitable job.6
.
Simplifying Equation 4.1 yields:
Pr(T < t) = 1 − e−λ(t)t
(
1 − F (wR))
, (4.2)
wherePr(T < t) means that the time when an agent finds a job,T , happens beforet. Of
course, if the solution for the reservation wage existed in closed form or could be observed,
estimation of Equation 4.2 would have been straightforward. However, many levels of recursion
and nonlinearities make it impractical for empirical applications.
Before going into explanation of the estimation methods it would be helpful to introduce fol-
lowing definitions and notations:Φ(t) = Pr(T < t). In duration analysis this is called a failure
function, which specifies the probability that the spell ”fails” before timet. In our case, the
”failure” is a change of state for a worker from ”unemployed”to ”employed”. The survival (or
survivor) function,S(t) = 1 − Φ(t), is the probability that the ”failure” does not occur before
time t. The hazard function,φ(t) =dΦ(t)/dt
S(t), is ”... the rate at which spells will be completed
at durationt, given that they last untilt” (Kiefer (1988, p. 651)).7 A tractable approach to tackle
the unemployment model empirically is to define the hazard function as:
φ(t) = λ(t)(
1 − F (wR))
. (4.3)
Then one must specify the functional relationship between the hazard function and the set of
exogenous parameters. A convenient starting point is the exponential specification. As Kiefer
puts it: ”The exponential [distribution] is simple to work with and to interpret, and is often an
adequate model for durations that do not exhibit much variation (in much the same way that
the linear regression model is simple and adequate if the data do not vary enough to reveal
important nonlinearities).” (Kiefer (1988, p. 552))8 Other specifications are considered in the
next sections: Section 4.1.1 describes the exponential andWeibull distributions, Section 4.1.2
gives an overview of the Cox method, Section 4.1.4 describesthe Kaplan-Meier product-limit
estimator.
36
4.1.1 Parametric Methods
Exponential Distribution
When the duration distribution is specified the model could be estimated by the method of
Maximum Likelihood. For an exponential distribution the hazard function can be given as:
φ(x, β) = ex′β (4.4)
The expected duration,E(t) is e−x′β. The log-likelihood function can be given as:
ln L =∑
dix′iβ −
∑
tiex′
iβ, (4.5)
whered = 0 if an observation is censored andd = 1 if it is uncensored. The respective
derivatives of the likelihood function are:
∂ ln L
∂β=
∑
dix′i −
∑
tiex′
iβx′i
∂2 ln L
∂β∂β ′= −
∑
tiex′
iβxix′i.
(4.6)
One could see that the log-likelihood function is concave, so numerical maximization is straight-
forward to apply.
Weibull Distribution
The Weibull model specifies the hazard function as:
φ(λ, t, β) = λα(λt)α−1. (4.7)
To facilitate estimation the following transformation would be useful:(tλ)α = eω. Then the
distribution function for the Weibull is:F (ω) = 1 − exp(−eω). Letting λ = exp(x′β) and
α = 1/σ:
ωi =ln ti − xiβ
σ. (4.8)
It is obvious from Equation 4.7 that the Weibull distribution is simply a generalization of the
exponential distribution, where the latter is a special case of the Weibull distribution when
α = 1.
37
The observed random variable is:
ln ti = σωi + xiβ. (4.9)
The log-likelihood can be given by:
ln L =∑
i
[
di
( ln ti − xiβ
σ− ln σ
)
− exp( ln ti − xiβ
σ
)]
, (4.10)
wheredi = 0 if an observation is censored anddi = 1 otherwise.
4.1.2 Proportional Hazard Specification and Semiparametric Estimation
Proportional hazard models, namely the Cox regression (developed by Cox(1972, 1975)), are
popular in econometrics of duration data as the baseline hazard needs not to be specified which
makes the model rather flexible. In this model the hazard function is specified as:
φ(t, x, β, φ0) = λ(x, β)φ0(t), (4.11)
whereφ0 is the ”baseline” hazard, corresponding toλ(x, β) = 1. Specifying the arrival rate as
λ(x, β) = exp(x′β), yields:
φ(t, x, β, φ0) = ex′βφ0(t). (4.12)
The algorithm of the Cox regression can be explained as follows. The completed durations are
ordered,t1 < t2 < . . . < tn9 the conditional probability that observation 1 ”fails” at duration
t1, given that any of then observations could have been concluded att1 isφ(t1, x1, β)
∑ni=1 φ(t1, xi, β) .
In the proportional hazard model this expression reduces to:
λ(x1, β)∑n
i=1 λ(xi, β), (4.13)
which 4.13 is the contribution of the shortest spell to the likelihood function. In the same
fashion, the contribution of thejth duration (given that the spells are ordered as described above)
isλ(xj , β)
∑ni=j λ(xi, β) . The likelihood function is then:
ln L(β) =
n∑
i=1
(
ln λ(xi, β) − ln[
n∑
j=1
λ(xj , β)])
. (4.14)
38
4.1.3 Competing Risks
Competing risks imply that there are several possible ways in which the spell can end or several
possible destination states from the state where the personor process dwells. For example,
in medical research the treatment of the disease may end withrecovery or death, in political
science the legislative term (for example in parliament) may end due to retirement or assumption
of another public office (be it a minister or a governor). In terms of the theoretical model
presented in Chapter the unemployed worker may become employed or withdraw from the
labor force. The conventional and most widely used method ofestimating the competing risk
models is to assume independence of risks. In this case, one first estimates the model treating the
failures due torisk 1as uncensored, considering the failures due torisk 2as censored. Secondly,
one estimates the model with failures due torisk 2 as uncensored, treating the failures due to
risk 1 as censored. This could seem quite strong an assumption. However, for conventional
estimation techniques theconditionalindependence suffices, i.e. independence is assumed after
having controlled for all the covariates. Put it another way, the errors in the first and the second
regression must be uncorrelated. If the assumption of independence of errors is fulfilled, the
competing risk model can be handled as a single risk model with other risks treated as censored
(see among others Katz (1986), Gonzalo and Saarela (2000), Qian and Correa (2003), Wolff
(2003), Wolff and Trubswetter (2003), Fitzenberger and Wilke (2004), Wolff (2004)).
The dependent competing risks are still a field with white spots as identification arises as a
problem, since once a person leaves his current state via risk i, then the duration until the
exit via risk j is not observed. Although much work has been done to identifythe model,
overall identifiability is still debatable. Hausman and Han(1990) devised methods to handle the
dependent competing risks. They estimated the full model with a bivariate error distribution.
They claim the model is identified only when there are as many continuous regressors as there
are competing risks. Their likelihood function had to be maximized with respect to exogenous
parametersβ1 andβ2, respective variancesσ1 andσ2, the coefficient of correlationρ, and the
unobserved mean durationst1 andt2.
One of the approaches to handle the competing risks is to allow the correlation of errors due to
unobserved heterogeneity. Hence, once unobserved heterogeneity is accounted for, the errors
are not correlated anymore (see Hougaard (1987) for an overview). These are also known as
frailty models. For example, using French data van den Berg, van Lomwel, and van Ours
(2003) estimated a model with independent errors but correlated heterogeneity terms. How-
ever, they could not reject the hypothesis of heterogeneityindependence. Gordon (2002) used
generalized dependent risk model of government office duration but also commented that the
generalized dependent risk model did not offer improvementover a stochastically independent
risk model.
39
4.1.4 Nonparametric Methods: Kaplan-Meier Product-Limit Estimator
Nonparametric methods does not use underlying theory or restrict the model to any specific dis-
tribution. They are more ”data-suited” than ”theory-suited”. However, graphical representation
of nonparametric estimates of survivor function could be a good starting point before going into
parametric methods as it may give some hints about the functional form of the hazard function.
For the sample ofn observations with no censoring the survivor function is:
S(t) = n−1. (4.15)
When censoring is present Equation 4.15 should be modified. We need to ordercompleted
(uncensored) durations in the sample in such a way thatt1 < t2 < . . . < tk (from the shortest to
the longest), withK < n if at least one observation is censored (the spell is not completed at the
calendar time the study terminates) or if there are ties (twoor more observations have the same
duration). Definehj as be the number of completed spells of durationtj, for j = 1, . . . , K. Let
mj be the number of censored observations betweentj andtj+1; and letnj be the number of
spells either completed or censored after durationtj .
nj =
K∑
i>j
(mi + hi). (4.16)
The hazardφ(tj) is the probability that a spell ends attj , given that it lasted untiltj . The
estimator forφ(tj) is:
φ(tj) = hj/nj, (4.17)
which is the number of ”failures” at durationtj divided by the number of ”potential failures” at
durationtj. The estimator of the survivor function is:
S(tj) =
j∏
i=1
ni − hi
ni
=
j∏
i=1
(1 − φi), (4.18)
which is called the Kaplan-Meier (due to Kaplan and Meier (1958)) orproduct-limitestimator.
”Essentially, this estimator is obtained by setting the estimated conditional probability of com-
pleting a spell attj equal to the observed relative frequency of completion attj” (Kiefer (1988,
p. 659)). One could see from Equation 4.18 that the estimatordoes not involve any explanatory
variables and does not account for heterogeneity of the data. Hence to apply the product-limit
estimation, the data should be split into homogenous groups.
40
4.2 The Kaplan-Meier Estimator and Withdrawals from the
Labor Market
Consider the next simulation scenario. I generated 1000 spells of 1000 unemployed individ-
uals all of whom eventually find employment, i.e. all spells are uncensored. The job finding
probability (I would call it the counterfactual here) is represented by the solid curve in Figure
4.1. Now suppose that for a group of workers who find jobs betweent = 508 andt = 585 the
potential search time,t∗ is 508. These workers have not found jobs untilt = 508, hence, they
withdraw from the labor market at this point in time (in this artificial data there are about 250
workers who have not found job untilt = 508 butwould have foundit at 508 < t < 585 if they
had searched longer). If we treat these observations as censored the estimated job finding prob-
ability would correspond to dashed curve in Figure 4.1 (Kaplan-Meier estimates). However, it
does not correspond to the true job finding probability. The real cumulative job finding proba-
bility is about 75 percent, since some 250 unemployed workers out of 1000 withdraw from the
labor market andneverbecome employed (represented by the dotted curve in Figure 4.1). But
if we look at the dashed line, the estimated probability equals one at highest value oft. This is
because duration models estimate the conditional job finding probability.
0.00
0.20
0.40
0.60
0.80
1.00
prob
abili
ty
0 100 200 300 400 500 600 700 800 900time
Figure 4.1: Job-finding Probability Simulation
Recall the Kaplan-Meier estimator of the survivor functiongiven in Equation 4.18. If an indi-
vidual withdraws from the labor market attj then attj this observation is treated as censored
and attj+1 this individual is no more ”at risk”. As a result the more individuals withdraw the
lower isnj at larget and hence lower the survival rate and higher the job finding probability.
If we do account for the dropouts, i.e. consider them ”at risk” then the job finding probability
41
would be represented by the dashed curve in Figure 4.1, whichin this case overestimates the
real share of job findings as it estimates theconditional job finding probability, in this case -
conditional upon staying in the labor market and not withdrawing. In a situation where dropouts
are not negligible it may be advisable to use the unconditional job finding probabilities as a basis
of comparison of the labor market performance.
To illustrate the idea consider the share of job findings at timet to be given as:
p(t) =Et
Ut + Et + Dt, (4.19)
whereEt denotes the number of workers who find jobs at timet, Ut - who remain unemployed,
andDt - who withdraw from the labor market. The survival rate at timeT is then given by:
S(T ) =
T∏
t=0
(1 − p(t)), (4.20)
and the job finding rate:1 − S(T ). However, if we estimate the duration model using conven-
tional methods the denominator in Equation 4.19 would beUt + Et as the withdrawn workers,
Dt, would not be considered ’at risk’.
When we estimate the duration model using the Kaplan-Meier method, the survival rate is given
by:
S(T )c =
T∏
t=0
(1 − p(t)c)), (4.21)
with p(t)c =Et
Ut + Etbeing a conditional probability of finding a job. The unconditional
probability of employment at timet can be given as:
p(t) = p(t)c(1 − p(t)d), (4.22)
wherep(t)d is the unconditional probability of withdrawing from the labor market. The uncon-
ditional dropout probability can be given as:
p(t)d = p(t)dc(1 − p(t)), (4.23)
with p(t)dc =Dt
Ut + Dt
being a conditional probability of dropout. It is easily verifiable that
solving the system of Equations 4.22 and 4.23 forp(t) would yield 4.19.
To find the unconditional job finding probability we would just need to recode the end of spells
for the withdrawing individuals tot = ∞, i.e. they never find jobs.10 For practical reasons it
suffices to replace the end of spell for the withdrawing individual with t = max(t).
42
For illustration I estimated conditional and unconditional job finding probabilities for unem-
ployed German workers. For empirical estimation the 1975-2001 data sample of the IABS
dataset has been used (description see in Appendix A.12 and Bender, Haas, and Klose (2000)).
The analysis is restricted to unemployed females. The comparison is drawn between four
groups: singles in West Germany, married in the West, singles in East Germany, and married in
the East. The data contain the information when the unemployment spells ended with an em-
ployment begin. As a proxy for withdrawals from the labor market the coding ”not available to
the labor market” has been used. If an unemployment spell is interrupted for less than 30 days
(during which the unemployed worker is not observed in the sample) and after that is coded
again as unemployed the two broken spells are considered as one continuous spell (see Fitzen-
berger and Wilke (2004)). If the spell ended in a way other than employment or withdrawal (for
example, the entitlement period for unemployment insurance benefits ended) it is considered as
censored. Naturally the unconditional job finding probabilities are lower than the conditional
ones. Figures 4.2 - 4.5 show the estimated conditional and unconditional probabilities.
0.2
.4.6
.8jo
b fin
ding
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure 4.2: Single females, West Germany. Solid line - conditional, dashed line - unconditional jobfinding probability.
The conditional and unconditional probabilities do not differ much with duration of unemploy-
ment less than a year. This implies that in the first year of unemployment the withdrawal rate is
negligible. However, after about one year of unemployment duration the discrepancy between
the conditional and unconditional probabilities grows as more and more unemployed withdraw
from the labor market. Comparing Figures 4.4 and 4.2 we couldsee that single females in West
Germany have higher employment prospects than their Eastern counterparts. After one year the
cumulative job finding probability is about 0.6 for females in the West and about 0.4 for females
43
0.2
.4.6
.8jo
b fin
ding
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure 4.3: Married females, West Germany. Solid line - conditional, dashed line - unconditional jobfinding probability.
in the East. If we look at Figures 4.5 and 4.3 we could see that the withdrawal rate for married
females in the East is greater than for their Western counterparts (this could be seen by a larger
gap between the estimated conditional and unconditional probabilities). Higher withdrawal rate
in the East indicates that the returns to search relative to the utility of nonparticipation are lower
in the East. This could be driven by many factors which can notbe answered here (lower wages,
lower arrival rate due to fewer vacancies, faster skill loss, higher utility of nonparticipation). If
we look at the conditional job finding probability for married females in East Germany, we get
an impression that after a year of unemployment almost a 30 percentage point improvement in
job finding probability is possible (from about 0.4 att = 365 to about 0.7 att = 900). However,
in reality (if we look at the unconditional job finding probability) after about 400 days of un-
employment no more improvement is possible. This implies that the increase in the cumulative
job finding probability after about 400 days is artificially driven by a diminishing number of
persons ”at risk”.
In the light of the theoretical model, the estimation results tell us that the potential search time,
t∗, for many female workers is about 400 days. After 400 days of unemployment their em-
ployment chances become low, the returns to search do not cover the search cost anymore, and
they withdraw from the labor market which results in a flat jobfinding probability aftert∗. This
could be for example driven by skill depreciation, which would be in accordance with findings
of Frijters and van der Klaauw (2006) who find that the fastestloss of skills occurs in the first
year of unemployment. If this is true, this could mean that after a year of unemployment skills
depreciate so much that employment chances become very low and unemployed workers quit
44
0.2
.4.6
.8jo
b fin
ding
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure 4.4: Single females, East Germany. Solid line - conditional, dashed line - unconditional jobfinding probability.
the labor market. This implies that the gain from active labor market policies that halt a skill
loss (or improve skills) is greater with an early intervention than with a late one.
45
0.2
.4.6
.8jo
b fin
ding
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure 4.5: Married females, East Germany. Solid line - conditional, dashed line - unconditional jobfinding probability.
46
Chapter 5
Spatial Search Theory and
Commuting
5.1 Introduction
Interregional mobility has long been regarded as an important adjustment mechanism equi-
librating regional disparities in wages and unemployment.Most of the works dealing with
mobility concentrate on migration thereby belittling the role of commuting in interregional in-
teraction. Interregional commuting, however, is an important process regulating the functioning
of regional labor markets in many modern economies. In Germany, for example, about 30% of
employed individuals in 1997 worked at locations other thantheir place of residence (with lo-
cations defined as NUTS-3 regions).
Another problem in studying the commuting process is that most of the works on commuting
are purely empirical (for example Raphael (1998)) or use theunilocational search models, i.e.
without explicitly allowing for search across locations (e.g. Eliasson, Lindgren, and Westerlund
(2003)). It seems natural to build commuting models on search theory.11 However, models
explaining search of the unemployed workers for jobs acrossspatial units are rare. Some of
them like Burda and Profit (1996) use the ”urn” principle in tradition of Pissarides (1979) where
the distribution of wage offers is degenerate (see Section 2.1.3). In this setting there is no
reservation wage property because all jobs are equally paid. The search strategy is then simple
- accept the first offer or do not participate at all. In Burda and Profit (1996), however, agents
optimize their search intensity.
Molho (2001) allows non-identical non-degenerate wage distributions at different locations and
exogenous separations. The similar approach with minor modifications has been adopted by
47
Damm and Rosholm (2003) and Arntz (2005). All of those modelsdeal with the search-migrate
decision. There are several works on commuting which are based on spatial search theoretical
models (see e.g. van Ommeren and van der Straaten (2005)). Unlike in the classical search
models, which assume wage dispersion, van Ommeren and van der Straaten (2005) allow for
constant wages but dispersion of distances to work. Unemployed individuals in their model
solve for maximal acceptable travel distance.
In this chapter, I allow for wage dispersion across regions and travel costs between regions.
Hence, reservation wage is not unique to an individual, i.e.agent sets different reservation
wages for the ”local” and ”distant” region. Moreover, I allow for arrival rate to be the function
of search intensity and therefore unemployed workers receive job offers only in those regions
where their search intensity is nonzero. In this setting themaximal acceptable travel distance is
determined by the condition of nonzero search intensity. Itwill be shown that exogenous factors
in the home region affect the reservation wage and search intensity in the ”distant” region and
vice versa. The model shows that the circle of possible commuting destinations is bounded
and the maximum distance a worker is willing to commute is determined by exogenous factors
in the model. It is also shown that the maximal travel distance condition involves a selection
mechanism which is further addressed in the empirical model.
Most importantly, I address in this chapter the problem of traditional search-theoretical ap-
proach when the wage offer distribution is asymmetric and dispersions in the two tails of the
distribution may vary independently of one another. The McCall model predicts increase in the
reservation wage with the mean and the mean-preserving spread of the wage offer distribution
(see for details Mortensen (1986)). However, the mean-preserving spread, i.e. changing the
spread holding the mean constant implies a symmetric stretching or compression of the wage
distribution which is problematic in the empirical context. If the wage distribution is not sym-
metric and variances in the left tail and in the right tail areallowed to change independently then
the mean-preserving spread is not an adequate measure anymore (asymmetric changes of the
dispersion in the left and right tail will change the mean as well). A good solution to this prob-
lem could be using median as a location parameter of the distribution and the median-preserving
spread in the left tail as a scale parameter for the left tail and the median-preserving spread in
the right tail as a scale parameter for the right tail of the wage distribution (see Moller and Al-
dashev (2006b) who first point to that problem). It will be shown that if the wage distribution
is not symmetric and variance in two tails of the wages distribution can change independently
of one another, the implications of the search theory slightly change. Namely, the dispersion
in the left tail of the wage distribution reduces reservation wage and search intensity, and the
dispersion in the right tail increases reservation wage andsearch intensity. As a consequence, in
my empirical model I include dispersion in the left tail and the right tail of the wage distribution
as separate additional regressors.
48
The numerical simulations show that increasing wages in theregion of origin and destination
by the same amount will increase reservation wages and search intensities in both regions.
Therefore, I use wages in destination and origin as separateregressors, not a ratio of wages
in the destination to the origin as it is usually done in the tradition of gravity models (see for
example Hatton and Williamson (2002), Mitchell and Pain (2003)).
5.2 Bilocational Search
Assume for simplicity that there are only two locations in the economy: place of residence and
a distant region (extension to a multilocational model is straightforward). Throughout the paper
I will use the terms regionA to denote the local labor market and regionB to denote the distant
labor market. I consider here only job search decisions of a resident of regionA as decisions
of residents ofB are derived in a likewise manner. Offers to work in regionA andB arrive to
the searcher according to a stationary Poisson process withthe arrival rateλA(θA) andλB(θB),
whereθA is the intensity with which a resident of regionA searches for jobs in the local labor
market andθB is the search intensity of a resident of regionA in distant labor market, which are
to be determined endogenously. The arrival rates satisfy the following properties:λ′A(θA) > 0;
λ′′A(θA) < 0 andλ′
B(θB) > 0; λ′′B(θB) < 0. Searching in each region involves a search cost
which is a function of search intensity. The cost functions satisfy the following properties:
c′A(θA) > 0; c′′A(θA) > 0 andc′B(θB) > 0; c′′B(θB) > 0.
The probability that an agent receivesn job offers in regionA during periodτ is given as:
qA(n, τ) =e−λA(θA)τ (λA(θA)τ)n
n!. (5.1)
In the same fashion, the probability that an agent receivesn job offers in regionB can be written
as:
qB(n, τ) =e−λB(θB)τ (λB(θB)τ)n
n!. (5.2)
The distributions of job offers in regionA andB are denoted asFA(w) andFB(w) respectively
and are exogenously given. If the searcher receives more than one job offer in both regions, he
picks the best one. LetG(w; n, m) be the distribution of maximalnet 12 wages fromn offers
drawn fromFA(w) andm from FB(w). The value of search for an individual residing in region
A can be given as:
49
Ω = (b − cA(θA) − cB(θB))τ + β(τ) × (5.3)[
∞∑
n=1
∞∑
m=1
qA(m, τ)qB(n, τ)
∫ ∞
0
max(W (w), Ω)dG(w; n, m)+
∞∑
n=1
qA(n, τ)qB(0, τ)
∫ ∞
0
max(W (w), Ω)dGA(w; n) +
∞∑
n=1
qB(n, τ)qA(0, τ)
∫ ∞
0
max(W (w) − δ/r, Ω)dGB(w; n) +
qB(0, τ)qA(0, τ)Ω]
,
whereβ(τ) is the discount factor which is equal toe−rt andδ is the cost of commuting per
period. The third term in 5.3 implies that if an agent does notget any offer in regionB (with
probabilityqB(0, τ), he picks the best offer (out ofn) in regionA (given that they exceed the
value of search), soGA(w; n) is the distribution of maximal wages fromn offers drawn from
FA(w). Using the same logic, the forth term means that if an agent does not get any offer
in regionA (with probabilityqA(0, τ)), he picks the best offer in regionB, soGB(w; n) is the
distribution of maximal wages fromn offers drawn fromFB(w). The second term in 5.3 implies
that if an agent receives more than one offer in each region, he picks the offer with the highest
wage. The last term in 5.3 means that the unemployed workers continue the search if they do
not receive any offers both in regionA andB.
It can be shown that in continuous time (τ → 0) the third term in 5.3 equals to zero and the
value of search equation simplifies to:
rΩ = b − cA(θA) − cB(θB) + λA(θA)∫ ∞
0max(W (w) − Ω, 0)dFA(w)+
λB(θB)∫ ∞
0max(W (w) − Ω − δ/r, 0)dFB(w).
(5.4)
It is straightforward to show thatrΩ = wRA = wR
B − δ, wherewRA andwR
B are reservation wages
in regionA andB respectively.
Proposition 1 Reservation wages inA (B) are decreasing (increasing) in the travel cost be-
tweenA andB. The corresponding elasticities are less than unity in absolute value.
Proof See Appendix
The interpretation of Proposition1 is straightforward: a higher travel cost reduces the value of
search so the reservation wage at locationA goes down, i.e. agents become less picky and
50
are ready to accept jobs they would not have taken before. Reservation wage inB cannot go
down with commuting cost, as part of the wage would have to be sacrificed to cover the travel
expenses. It is shown, however, that0 6∂wR
B
∂δ6 1, so the elasticity is less than unity in
absolute value, which means that although the wage aspirations become higher at locationB,
the net demanded wage is lower than before, so, indeed, agents are less picky.
It is interesting to see the effects of changes in the momentsof the wage offer distribution on
reservation wages. It is common in the search literature to use the mean and the mean-preserving
spread to characterize the wage distribution due Rotschildand Stiglitz (1970). However, the
classical Rotschild-Stiglitz definition of the mean-preserving spread says that a distributionF2
with a higher variance is a mean-preserving spread of the distributionF1 if they have the same
mean and∫ w
0
F2(x)dx >
∫ w
0
F1(x)dx.13 (5.5)
The classical definition gives unambiguous results that a higher mean-preserving spread leads
to higher reservation wages. However, the Rotschild-Stiglitz definition is impaired with much
inflexibility as it implies a symmetric stretching or compression of the distribution, which is
very unlikely to be observed empirically. If one tries to change the spread asymmetrically
whilst preserving the mean then the results are not in accordwith the theory.
Consider an example below.
Figure 5.1: Mean-preserving spread. An example
x
1.8751.6251.3751.125
0.5
0.875
0
1
0
1.5
1.75
1
x
1
0
1.51.25
1.5
2
0.5
both distributions have the mean 1.4375 and the variance 0.079427
Call the distribution to the left on the figure 5.1F1 and the distribution to the rightF2. Both
distribution have the same mean and the variance:E(x) = 1.4375 and var(x) = 0.079427.
The reservation wage function in a simple unilocational context is given as:wR = b − c +
51
λr
∫ ∞
wR(w − wR)dF (w). Let b − c = 0 andλ/r = 6 without the loss of generality. Then the
reservation wage of the individual facing the distributionF1, wR1 = 1.28324 and the reservation
wage of the person facingF2, wR2 = 1.27653. So despite having the same mean and variance,
these two distributions cause different reservation wages. Ambiguity comes from the fact that
the condition 5.5 does not hold.
As an alternative to Rotschild-Stiglitz mean-preserving spread I propose using the median-
preserving spread. To be more precise, I suggest using the median as a location parameter of
the distribution and the median-preserving spread in the left tail as a scale parameter of the left
tail and the median-preserving spread in the right tail as a scale parameter of the right tail of the
wage distribution (see also Moller and Aldashev (2006b) who address this issue).
Define the median of the wage offer distribution asw, the median-preserving spread in the
right tail asσR, and the median-preserving spread in the left tail asσL. The spread is median-
preserving if for any arbitraryσR1 < σR2 andσL1 < σL2:
F (w; σR1) = F (w; σR2) = F (w; σL1) = F (w; σL2) = 1/2. (5.6)
Moreover,
∫ w
0
F (w; σL1) dw <
∫ w
0
F (w; σL2) dw;
∫ ∞
w
F (w; σR1) dw >
∫ ∞
w
F (w; σR2) dw. (5.7)
Proposition 2 The reservation wage for regionA increases with the median wage of regionA
and regionB (but the elasticity is less than unity) and the median-preserving spread in the right
tail of the wage distribution of regionA and regionB. It decreases with the median-preserving
spread in the left tail of the wage distribution of regionA and regionB. The same applies for
the reservation wage for regionB.
Proof See Appendix.
Corollary The reservation wage set by a searcher for any location does not only depend on
wages in this location, but also on wages in all other locations.
Some ideas in Proposition 2 are not new. For example in classical search models the elasticity
of the mean wage is also positive and less than unity. However, in standard search models
reservation wage increases with the variance. This is because ”. . . the worker has the option of
waiting for an offer in the upper tail of the wage distribution” (Mortensen (1986, p. 865)). The
effect of the median-preserving spread in the right tail hasthe same interpretation. However,
increasing the spread in the lower tail of the wage distribution allocates more probability mass
52
to the jobs with lower wages and less probability mass to the jobs with higher wages. Moreover,
some of the probability mass has gone to jobs which pay wages below the reservation wage.
To compensate for this loss of probability mass, reservation wage declines with the median-
preserving spread in the left tail.
It is assumed that agents optimize their search effort to maximize the returns to search. This
implies that search intensity in the regionA solves:
c′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w), (5.8)
and likewise:
c′B(θB) =λ′
B(θB)
r
∫ ∞
wRB
(w − wRB)dFB(w). (5.9)
Proposition 3 Higher commuting costs make agents search harder in the local labor market
and less intensively in a distant labor market.
Proof See Appendix.
The result of Proposition 3 is important that it establishesinterdependency of search intensi-
ties. When commuting costs go up, the net expected wage in a distant labor market decreases
and wealth-maximizing agents not just reduce their search intensity in the distant region, they
reallocatethis search intensity to search activity in the local labor market.
Proposition 4 Agents search harder in the local labor market and less intensively in the distant
labor market if median wage or the median-preserving spreadin the right tail of the wage
distribution increase in the home region or when the median-preserving spread in the left tail
of the wage distribution decrease in the home region. If median wage or the median-preserving
spread in the right tail of the wage distribution increase both in the distant and local labor
markets by the same amount, agents search harder in both regions; and if the median-preserving
spread in the left tail of the wage distribution increase both in the distant and local labor
markets by the same amount, agents search less intensively in both regions.
Proof See Appendix.
As in Proposition 3 the result establishes reallocation of search intensity to regions where ex-
pected wage increases. An important result of Proposition 4is also that if the median wage in
both regions increases by the same amount, search intensities increase in both regions.
53
5.3 Maximal Acceptable Travel Cost
The next important issue which I would like to address in thischapter is the maximal acceptable
commuting cost. The necessary condition that a resident of regionA searches in regionB is
that the returns to search in a distant labor market cover thesearch costs. It is then possible that
after some critical level of commuting cost the returns to search do not cover the search costs
anymore. The condition that the returns to search are fully offset by the search costs is called
here azero search conditionmeaning that at this point, a resident ofA is indifferent between
searching in both regions or in the local labor market only. It is possible to determine the value
of the travel cost making the searcher indifferent between investing in search in regionB, or
search inA only. So far we have a system of four equations with four unknowns:
wRA = b − cA(θA) − cB(θB)+
λA(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w) +
λB(θB)
r
∫ ∞
wRB
(w − wRB)dFB(w)
wRA + δ = wR
B
c′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w)
c′B(θB) =λ′
B(θB)
r
∫ ∞
wRB
(w − wRB)dFB(w).
(5.10)
Solving this system we can find reservation wages inA andB and search intensities inA and
B. In order to find the travel cost which makes an agent indifferent between searching inA and
B and searching inA only, the following condition has to be imposed:
cB(θB) 6λB(θB)
r
∫ ∞
wRB
(w − wRB)dFB(w). (5.11)
Restriction in 5.11 sets the condition that the returns to search in regionB should not be less
than the costs of the search inB (the Kuhn-Tucker condition). Solving Equation 5.10 and 5.11
simultaneously in five endogenous variables:wRA, wR
B, θA, θB, δ yields the value of the maximal
acceptable commuting cost. If the travel costs exceeds thiscritical level, then the returns to
search in a distant region do not cover search costs. Hence, aresident ofA will invest in search
only in those regions which lie within the acceptable travelcost. If for the ease of exposition we
assume that the commuting cost depends only on the physical distance between regions, then
Equation 5.10 and 5.11 enable us to find the maximal radius of possible commuting destinations.
This is schematically shown in Figure 5.2.
54
A
B
B
A
Figure 5.2: Circle of possible commuting destinations
In Figure 5.2 to the left we see that the circle of acceptable commuting distance around regionA
(small circle) does not include regionB, hence, a searcher in regionA does not commute toB.
But the situation changes, if, for example, wages in both regions increase by the same amount
as in Figure A.3, the circle inflates (as on the graph to the right) and a searcher in regionA may
commute toB. The reverse is also true, i.e. regionB is an acceptable commuting destination
for a resident ofA (situation on the right graph). But if the wages in both regions decrease the
circle shrinks (graph to the left) andB is no more a possible commuting destination.
5.4 Participation
The probability that a resident of regionA commutes toB can be given by the hazard rate
λ(θB)(1 − F (wRB)) (probability that he is offered a job inB and the offered wage exceeds his
reservation wage). The commuter flow fromA to B is SA · λ(θB)(1 − F (wRB), whereSA is
the number of active searchers who live in regionA. If workers are homogenous the number of
active searchers is constant (either all or none participate in the labor market). If wages in both
regions go up, the agents would search harder in regionB and, hence, the commuter flow would
grow. At the same time the probability of a match in the regionof residence also increases. This
is in contrast to the approach used in gravity models where the ratio of average wages inA and
B is used. This approach would imply that only changes in theratio would affect the commuter
flow and if this ratio does not change the commuter flow also remains unchanged. Proposition 4
shows that equal growth of wages in regionsA andB increases the commuter stream between
A andB although the ratio of median wages may not necessarily change.
Now suppose that workers differ in the value they attach to leisure (see also Albrecht and Axell
(1984), Moller and Aldashev (2006b)). As in Section 4.1.3,I allow for three states: employ-
ment, unemployment, and nonparticipation. Individuals donot participate in the labor market
if their returns to search are less than the value of not participating in the labor market. Sup-
55
pose that individuals can stay inactive thereby earning pure leisure which is worthb. If they
participate, their reservation wage as previously defined is:
wRA = b− cA(θA) − cB(θB) + λA(θA)
∫ ∞
wRA
(w − wRA)dFA(w) + λB(θB)
∫ ∞
wRB
(w − wRB)dFB(w).
(5.12)
The participation condition is thus:wRA > b. Ruling out corner solutions, there exists a
”marginal individual” for whomwRA = b. Suppose the values of leisure are distributed across
individuals with the distribution functionG(b). The participation rate is thenG(b), whereb is
defined asb ≡ wRA(b). It can be shown that the participation rate increases with the median
wage (both in origin and destination), median-preserving spread in the right tail (both in origin
and destination), but decreases with the median-preserving spread in the left tail (both in origin
and destination).
Suppose that the worker with the lowest value of leisure hasb = 0. 14 The number of com-
muters from regionA to B can be given as:
∫ b
0
λB(θB)(
1 − FB(wRB)
)
dG(b)POPA, (5.13)
wherePOPA stands for the population size of regionA. Increase in the median wage in the
destination would increaseb. Moreover, increase in the median in regionB would make agents
reallocate their intensity to regionB (see Section 5.2) and thereforeλB(θB) also increases.
The reservation wage inB increases with the median wage inB but the elasticity is less than
unity, hence,(
1 − FB(wRB)
)
also increases with the median wage inB. As a consequence, the
commuter flow fromA to B unambigously increases with the median wage in the destination.
If the median wage in the origin increases, the participation rate also goes up (b increases).
However, agents would reallocate their search intensity toregionA and hence,λB(θB) declines.
This implies that when wages increase in the origin, less commuting is possible because agents
start searching harder in the origin and less harder in the destination, but on the other hand, more
commuting is possible because overall number of searchers in the origin increases. Hence, the
overall effect is ambiguous.
The effect of spreads cannot be analytically determined. For example, increase in the spread
in the right tail of the distribution in the destination would increase participation (b is higher)
and search intensity (λB(θB) would increase). However, the term1 − FB(wRB) might decrease
with the spread in the right tail (see proof in Mortensen (1986) on the ambiguity of the effect
of dispersion). The effect of the increase in the spread in the right tail of the distribution in the
56
origin is even less clear asb andλB(θB) will move in the opposite directions with the changes
of the spread in the origin.
57
Chapter 6
Empirical Estimation of a
Commuting Model
6.1 Introduction
Consider residents of regionA looking for a job and firms located inB with vacancies. If
these workers fromA are matched with vacancies inB (given that residents ofA search in
B with positive search intensity) they commute. Hence, we could treat the commuter stream
from A to B as successful matches of searchers from regionA with vacancies in regionB. The
expected number of commuters fromA to B, as already discussed in Chapter 5 can be obtained
asSA · λB(θB)(1 − F (wRB)). Search intensity and reservation wage of a resident of region
A searching in regionB depend on characteristics of both origin and destination regions: the
arrival rate, search cost, median wage, median-preservingspread in the left and right tail, value
of leisure (and if we extend it to a multi-region model then also on characteristics of all other
regions).
For empirical convenience the matching function can be taken as Cobb-Douglas which in prin-
ciple can be estimated by OLS after taking logs. This approach is followed by Gorter and van
Ours (1994) and Burda and Profit (1996) who analyze matching of unemployed workers with
open vacancies. However, in case of commuting, ”. . . the predominance of zeros and the small
values and clearly discrete nature of the dependent variable suggest that we can improve on least
squares and the linear model with a specification that accounts for these characteristics” (Green
(2003, p. 740)). Many works revert to discrete probability process when analyzing commuting
streams (see among others Raphael (1998), Flowerdew and Aitkin (1982), Guy (1987), Yun
and Sen (1994)). Although it has to be noted that classical gravity models are still to be found
58
in some works (e.g. Foot and Milne (1984), Hatton and Williamson (2002), Mitchell and Pain
(2003)).
6.2 Data and Descriptive Evidence
The commuter stream data used in this paper are produced by the Institute for Labor Research
(IAB) from the employment register of the Federal Labor Office with regional information. The
data contain the flows of commuters in 1997 between 440 NUTS-3regions, which makes 193
160 observations. Unfortunately, the data do not differentiate commuters with respect to gender.
The dependent variable is then the commuter flow from regioni to j. Wage quantiles of the
wage distribution were calculated using the IABS-REG microdataset for 1997 (see description
of the data in Appendix A.12 and Bender, Haas, and Klose (2000)). Data on population of
NUTS-3 regions were taken from the INKAR database of the Federal Office for Building and
Regional Planning. The information on travel time was takenfrom the data of the Institute for
Regional Planning of the University of Dortmund (see description of the data in Appendix A.12
and Spiekermann, Lemke, and Schurmann (2000)).
The exogenous parameters used for estimation are:15
• wi - the median log wage in regioni,
• wj - the median log wage in regionj,
• POPi - log population of regioni (as an indicator of the size of the labor market ini),
• POPj - log population of regionj,
• D8/D5i - the log difference of 8th to 5th decile of the wage distribution in regioni (as
an indicator of the median-preserving spread in the right tail),
• D8/D5j - the log difference of 8th to 5th decile of the wage distribution in regionj,
• D5/D2i - the log difference of 5th to 2nd decile of the wage distribution in regioni (as
an indicator of the median-preserving spread in the left tail),
• D5/D2j - the log difference of 5th to 2nd decile of the wage distribution in regionj,
• tij - log travel time between regionsi andj.
Since in a multi-region model commuting streams depend on exogenous parameters of all other
regions, I also include average wage in other regions (except i andj) with inverse travel time
59
as weights and average population in other regions (excepti andj) with inverse travel time as
weights. These spatially weighted variables are calculated as:
Wij =∑
s 6=i,s 6=j
ws
tisand Pij =
∑
s 6=i,s 6=j
POPs
tis. (6.1)
Table A.2 presents the descriptive statistics of the data used. The average travel time between
pairs of regions is about 4.5 hours. However, the distribution of travel times is rather dispersed.
The same can be said about the distribution of population.
Table A.3 presents the descriptive statistics separately for regions of West Germany and East
Germany. One could see that average daily earnings are substantially higher in the West, but
on the other hand, more dispersed. Western regions are also on average larger in population
size, however, with very high variation of population size.Eastern regions experience higher
unemployment rates. Moreover, Eastern regions are on average farther from each other in terms
of travel time than the Western regions. However, the dispersion of travel times in the East is
also larger.
This stresses importance of controlling for differences insearch behavior in East Germany and
West Germany. Figure 6.1 shows the distributions of commuting flows from West Germany
(left panel) and East Germany (right panel). The figure showsthat in West Germany the share
of stayers is larger than in the East which is reflected by a larger probability of zero commuting.
The differences with respect to destination are also important. As shown in Figure 6.2, the
share of zeros for destination East Germany is larger than for the West, which tells us that job
searchers are more likely not to commute to East Germany. Figure 6.3 shows that movements
from the East to the West are more likely than commuter movements from West to the East.
To account for these differences, I introduced two dummy variables indicating whether the
region of origin is East or West Germany and whether the destination is East or West Germany.
60
0.2
.4
0 10 20 30 0 10 20 30
0 1
Den
sity
commuting flowGraphs by east_or
Figure 6.1: Distribution of commuting flows. Origin West Germany (left)and origin East Germany(right).
0.5
0 10 20 30 0 10 20 30
0 1
Den
sity
commuting flowGraphs by east_d
Figure 6.2: Distribution of commuting flows. Destination West Germany (left) and destination EastGermany (right).
61
0.2
.4.6
0.2
.4.6
0 10 20 30 0 10 20 30
0, 0 0, 1
1, 0 1, 1
Den
sity
commuting flowGraphs by east_or and east_d
Figure 6.3: Distribution of commuting flows. Origin and destination West Germany (upper left panel),origin West, destination East Germany (upper right panel),origin East, destination West(lower left panel), origin and destination East (lower right panel).
6.3 Clustering and Robust Variance Estimation
The vector of commuter flows has the following form:
A B
A C
A D
A . . .
... . . .
B A
B C
B D
B . . .
... . . .
(6.2)
62
The first column represents an indicator for the region of origin, the second column is an indi-
cator for the destination. After some permutations 6.2 can be written as:
B A
C A
D A
. . . A
... . . .
A B
C B
D B
. . . B
... . . .
(6.3)
The above shown representation illustrates that the data are clustered with respect to the origin
and the destination, which due to omitted region-specific effects results in correlation of errors
within clusters and, hence, biased standard errors. This iscalled Moulton’s bias due to Moulton
(1990). This can be easily demonstrated on the following example. Denoteyir as the dependent
variable;i is an index for an observation andr is an index for a region. Let the equation of
interest be of the form:
yir = xirβ + υr + ǫir. (6.4)
If for a regionr we have several observations and the termυr is unobserved, then Equation 6.4
can be written as:
yir = xirβ + uir, (6.5)
with a compound error termuir = υr + ǫir. Error terms within each region would be correlated
because they would include the same termυr. Therefore, in all further estimations I control for
error correlation within clusters and report corrected standard errors (more on clustering and
error correlation see in Moulton (1990), Dickens (1990), Blanchflower and Oswald (1995)).
Estimation of the correct variance in the presence of clustering is as follows. Define the score
function as:S =∂L
∂(xβ), whereL is the log-likelihood function,x is the matrix of independent
variables, andβ is the coefficient vector. The Hessian is defined as:H =∂2L
∂(xβ)2. Suppose
we haveR regions within which observations are clustered. The robust standard errors can be
calculated as:
var(β) = D−1(
R∑
r=1
U ′rUr
)
D−1, (6.6)
63
whereD =∑R
r=1
∑
i∈r(Hirx′irxir) andUr =
∑
i∈r xirSir, which is a contribution of each
cluster to the score.
Consider a simple case with two clustersA andB with two observations per cluster. Let the
vector of explanatory variable be of the form:x1A, x2A, x3B, x4B. If we ignore clustering and
consider each observation as independent, the term∑R
r=1 U ′rUr would simplify to
∑
(xiSi)2 =
(x1AS1A)2 + (x2AS2A)2 + (x3BS3B)2 + (x4BS4B)2. If we take into account clustering then∑R
r=1 U ′rUr = (x1AS1A + x2AS2A)2 + (x3BS3B + x4BS4B)2, which is larger. This simple
illustration shows that ignoring correlation of errors within clusters causes a downward bias in
the variance of the estimated coefficients and in many cases coefficients of variables appear to
be significant when in reality they are not.
6.4 Estimating the Poisson Model
In its simplest form, the discrete probability process of a variable can be modeled by the Poisson
model:
Pr(Yij = yij) =e−λijλ
yij
ij
yij!. (6.7)
Most commonlyλij is parametrically specified as:λij = exβ with x as a matrix of exogenous
variables. The problem of the Poisson model is that in the Poisson distribution the mean equals
the variance, which in case of overdispersion, i.e. if the variance is in reality greater than the
mean, causes a downward bias in the standard errors. More general negative binomial model
is used instead of Poisson regression by many authors (the negative binomial specification is
discussed in Section 6.5).
64
Table 6.1: Estimation results of the Poisson model (Model 1 includes two dispersion parameters asregressors, Model 2 only one). Dependent variable - commuting flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
East origin −0.12 0.15 0.01 0.16
East destination 0.43 0.17 0.36 0.17
log travel time −2.50 0.03 −2.49 0.03
log median wage (origin) −3.00 0.61 −3.09 0.71
log D5/D2 (origin) 5.74 0.91 - -
log D8/D5 (origin) −6.93 1.48 - -
log D8/D2 (origin) - - 8.63 2.52
log median wage (destination) 4.09 0.54 3.91 0.60
log D5/D2 (destination) −2.42 1.03 - -
log D8/D5 (destination) 2.27 1.72 - -
log D8/D2 (destination) - - −4.32 3.26
log POP (origin) 0.80 0.08 0.79 0.09
log POP (destination) 0.89 0.06 0.89 0.07
spatially weighted POP (×100) −0.40 0.08 −0.37 0.09
spatially weighted median wage −0.04 0.03 −0.05 0.04
const 3.85 3.39 −0.02 6.42
R2 0.82 0.83
N 190 532
Coefficients significant at least at 5% level are in bold.
Using the results obtained in Section 5.4 we could make certain predictions about the sign
of the effects of different variables. For example, the effect of the median wage of the wage
distribution in the destination should have positive effect on commuting, whereas the effect
of the median wage in the origin is ambiguous. The travel timeis expected to have negative
influence on commuting. The surrounding regions might also attract searchers to them thereby
distracting commuting from destination region. This is usually called intervening opportunities
(see the classical work of Stouffer (1940)). Considering that regions with higher wages attract
more,Wij , ceteris paribus, should have a negative effect on commuting. The effect of dispersion
parameters is ambiguous.
Table 6.1 present the results of the Poisson model. The modelfits the data well which is shown
by a highR2 value. East dummy is significant and positive. This tells us that controlled for
other factors commuter streams to East German regions are by40% larger than to the West.
65
This might be partly explained by higher job availability inthe West, so for the Western workers
it is easier to find a job in the region of residence (although it is not possible to either prove or
disprove that within this model). Travel time as expected isnegative and highly significant.
One percentage increase in the travel time reduces the commuter stream by 2.5 percent. The
moments of the wage distribution in the origin are highly significant and have the expected sign.
Increase in the median wage in the origin by one percent reduces the number of commuters by
three percent. In the light of the theoretical model this implies that the negative effect of the
reallocation of the search intensity from the destination to the origin (the agents reallocate their
search intensity and search harder in the region of origin and less intensively in the destination)
dominates over the positive effect of the wage increase on participation. Increase in the spread
in the right tail of the distribution (proxied in the model bythe log ratio of the eighth to the
second decile) also reduces search in the destination and hence leads to fewer commuters. The
spread in the left tail of the wage distribution in the origin(proxied in the model by the log ratio
of the eighth to the second decile) increases the outflow of commuters. Increasing the median
wage in the destination by one percent would result in four percentage increase of the commuter
flow. It is plausible that the coefficient for the effect of themedian wage in the destination is
greater in magnitude than the coefficient for the origin as participation and search intensity in
the destination move in the same direction with the change ofthe median in the destination, and
participation and search intensity in the origin move in theopposite directions with the change
of the wages in the origin.
The effects of wages in the origin and destination on commuting enable us to calculate the net
commuting effect. If we have two regions,A andB and median wages are only explanatory
variables, we could write:
yAB = wAβ1 + wBβ2
yBA = wBβ1 + wAβ2.
(6.8)
Equation 6.8 shows that if, for example, the median wage in regionA increases, the commuter
flow from A to B is reduced byβ1 and on the other hand, the commuter flow fromB to A
increases byβ2. So the effect of an increase in the median wage inA on the commuter saldo
betweenA andB (yAB − yBA) according to the results of the Poisson model is:0.97yAB −1.04yBA. If the original commuter saldo is zero (yAB − yBA = 0), then increase of the median
wage in regionA by one percent would reduce the commuter saldo of the regionA by 7 percent.
The effects of the population size in the origin and destination are positive and significant (al-
though elasticity is less than unity) which implies that larger regions interact more with each
other. The effect of population size on the commuter saldo isnegligible (if the original com-
66
muter saldo is zero). The effect of the population size in thesurrounding regions other than
destination reduces the commuter flow. This implies that regions that are closer and larger are
more likely to be intervening opportunities for potential commuters. The effect of an average
wage at locations other than the region of destination is statistically insignificant.
For comparison in Table 6.1 estimates of the Poisson regression are presented where instead of
two separate parameters for spreads in the tails, only one spread parameter (the ratio of8th to
2nd decile) is used.
6.5 Estimating the Negative Binomial Model
Negative binomial can be obtained from the Poisson model by introducing a random component
into λ. To be more precise letλij = µijνij. If we specifyµij = exβ andνij follow a Gamma
distribution withE(νij) = 1 and var(νij) = α, then the distribution ofyij can be written as:
h(yij, α, µij) =Γ(α−1 + yij)
Γ(α−1)Γ(1 + yij)
(
α−1
α−1 + µij
)α−1(
µij
α−1 + µij
)yij
. (6.9)
The first two moments of the negative binomial distribution are: E(yij) = µij and var(yij) =
µij(1 + α · µij). If α is zero thenE(yij) = var(yij) and negative binomial is identical to the
Poisson. Hence, testingα = 0 after estimating the negative binomial is identical to testing the
negative binomial specification vs. the Poisson.
67
Table 6.2: Estimation results of the negative binomial model (Model 1 includes two dispersionparameters as regressors, Model 2 only one). Dependent variable - commuting flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
East origin 0.62 0.07 0.59 0.07
East destination 0.75 0.14 0.62 0.14
log travel time −2.67 0.02 −2.66 0.02
log median wage (origin) −0.07 0.21 −0.01 0.23
log D5/D2 (origin) 0.82 0.39 - -
log D8/D5 (origin) 0.44 0.60 - -
log D8/D2 (origin) - - 2.54 1.13
log median wage (destination) 3.99 0.45 4.35 0.51
log D5/D2 (destination) −0.90 0.83 - -
log D8/D5 (destination) 7.41 1.39 - -
log D8/D2 (destination) - - 5.71 2.31
log POP (origin) 1.01 0.03 1.01 0.03
log POP (destination) 0.98 0.05 0.99 0.06
spatially weighted POP (×100) −0.14 0.03 −0.14 0.03
spatially weighted median wage −0.13 0.01 −0.13 0.01
const −13.35 1.19 −22.55 2.28
α 1.24 0.02 1.26 0.02
N 190 532
Coefficients significant at least at 5% level are in bold.
Table 6.2 present the results of the negative binomial model. The termα is significant, which
implies that the negative binomial specification performs superior to the Poisson. The East
dummies for both origin and destination are significant and positive, which means that com-
muting is more intensive between East German regions. The median wage in the origin, the
spread in the right tail of the wage distribution in the origin and the spread in the left tail in the
destination appear to be insignificant in this specification. The effect of the spread in the left
tail in the origin is significant, however, the magnitude is substantially reduced. The magnitude
of the coefficient for the spread in the right tail in the destination on the other hand more than
tripled. The effect of the median wage in the destination remains robust. Increasing the median
wage in the destination by one percent would result in about four percentage increase of the
commuter flow, which is not different from the results of the Poisson model.
One percentage increase in the travel time would reduce the commuter flow by 2.7 percent,
68
which is not much different from the Poisson model estimate (2.5 percent). The effects of
the population size in the origin and destination are positive and significant with unit elasticity
which implies that larger regions interact more with each other. The effect of population size
on the commuter saldo is negligible (if the original commuter saldo is zero). Larger popula-
tion in the surrounding regions other than destination reduces the commuter flow (although the
magnitude is smaller than in the Poisson model). This implies that regions that are closer and
larger are more likely to be intervening opportunities for potential commuters. The effect of an
average wage at locations other than the region of destination is statistically significant unlike
in the Poisson model. This implies that regions with higher wages are more likely to attract
potential commuters thereby reducing the commuter flow to other destinations.
For comparison in Table 6.2 estimates of the negative binomial regression where instead of two
separate parameters for spreads in the tails, only one spread parameter (the ratio of8th to 2nd
decile) is used are also presented.
6.6 Zero Inflated Models
If we look at the distribution of commuter flows we see that about one third are zeros (see
for example Figures 6.1 - 6.3). This should not be surprising. In the theoretical model the
zero search intensity condition implies that if a region lies beyond the circle of acceptable
commuting destinations, agents do not search there. This isplausible as the travel time between
some regions is about 8-10 hours, making commuting virtually impossible. We could split the
decision making process of a resident ofA into two stages. At stage one, he solves for the
maximal acceptable travel cost (given exogenous variablesof all regions) and if a region lies
within the acceptable travel cost he allocates his effort into search in this region, i.e. enters stage
two. At stage two, he searches for jobs at this location and, if successful, becomes consequently
employed. This implies zero and nonzero commuting streams are generated by two different
processes. Zeros could mean that agents do not search in these regions - decision taken by
workers only, achoiceoutcome. Positive outcomes are generated through a different process
- matching, arandomoutcome. To handle models like this a class of hurdle modes has been
developed, for example the zero-inflated negative binomialor zero-inflated Poisson approach.
In these models zeros and positive outcomes are generated bydifferent processes. Zeros have
the densityh1(·), soPr(yij = 0) = h1(0). Positive outcomes come from the truncated density
h2(yij|yij > 0) =h2(yij)
1 − h2(0). The probability thatyij is drawn from this truncated density is
1 − h1(0). Hence:
69
q(yij) =
h1(0) if yij = 0
h2(yij)
1 − h2(0)(1 − h1(0)) if yij > 0
(6.10)
The likelihood function follows immediately from Equation6.10.
Testing the zero-inflated Poisson vs. Poisson and zero-inflated negative binomial vs. negative
binomial model is not trivial as the models are non-nested.16 The Vuong test for non-nested
hypotheses can be applied in this setting. Let the predictedprobability that the random variable
Y equalsyi in the standard negative binomial (or in the Poisson) model be f2(yi|xi) and in the
zero-inflated negative binomial (or in the zero-inflated Poisson) –f1(yi|xi). We could estimate
both models and obtain the log-likelihood functions,L2 andL1. Denotemi = ln(f1(yi|xi)
f2(yi|xi)
)
.
The null hypothesis of the Vuong test is:
H0 : E(mi) = 0, (6.11)
which implies that the two specifications are equally close to the true model. Vuong shows that
under general conditions:
1
nLR
a.s.−→ E(mi), (6.12)
whereLR = L1 − L2 is the likelihood ratio test. Vuong demonstrates that:
ν ≡ LR√nω
D−→ N(0, 1), (6.13)
whereω =1
n
n∑
i=1
m2i −
(1
n
n∑
i=1
mi
)2
.
The result implies thatν has a limiting standard normal distribution. Hence, ifν > 1.96, the
test favors model 1, i.e. zero inflated model. Ifν < −1.96 , the test favors the non-inflated
model at 5% significance level. If−1.96 < ν < 1.96, the test favors neither model. The
Vuong test is intuitively appealing. If the null hypothesisis true, the log-likelihood ratio should
be on average zero. Iff1 is the correct specification then on average the log-likelihood ratio
should be significantly greater than zero. Iff2 is the correct model, the log-likelihood ratio
should be significantly less than zero. Hence, in principle,the Vuong test is simply theaverage
log-likelihood ratio suitably normalized (see also Clark and Signorino (2003)).
70
Table 6.3: Estimation results of the zero-inflated Poisson model (Model 1 includes two dispersion
parameters as regressors, Model 2 only one). Dependent variable - commuting flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
East origin −0.15 0.15 −0.02 0.16
East destination 0.44 0.17 0.36 0.17
log travel time −2.46 0.03 −2.45 0.03
log median wage (origin) −2.89 0.59 −2.97 0.68
log D5/D2 (origin) 5.65 0.89 - -
log D8/D5 (origin) −6.69 1.46 - -
log D8/D2 (origin) - - 8.60 2.47
log median wage (destination) 4.01 0.54 3.82 0.59
log D5/D2 (destination) −2.58 1.03 - -
log D8/D5 (destination) 2.19 1.71 - -
log D8/D2 (destination) - - −4.81 3.24
log POP (origin) 0.79 0.08 0.77 0.08
log POP (destination) 0.87 0.06 0.87 0.06
spatially weighted POP (×100) −0.41 0.08 −0.38 0.09
spatially weighted median wage −0.03 0.03 −0.05 0.04
const 3.84 3.31 0.49 6.30
inflate
East origin −1.08 0.12 −0.98 0.12
East destination −0.74 0.20 −0.57 0.19
log travel time 1.69 0.06 1.71 0.06
log median wage (origin) −1.66 0.51 −1.88 0.54
log D5/D2 (origin) 0.90 0.68 - -
log D8/D5 (origin) −5.62 1.19 - -
continued. . .
71
Table 6.3: Estimation results of the zero-inflated Poisson model (Model 1 includes two dispersion
parameters as regressors, Model 2 only one). Dependent variable - commuting flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
log D8/D2 (origin) - - −3.16∗
1.82
log median wage (destination) −3.40 0.68 −3.80 0.29
log D5/D2 (destination) −1.61 1.22 - -
log D8/D5 (destination) −9.62 1.89 - -
log D8/D2 (destination) - - −14.48 0.72
log POP (origin) −1.13 0.06 −1.14 3.44
log POP (destination) −1.17 0.07 −1.18 0.07
spatially weighted POP (×100) −0.30 0.10 −0.27 0.10
spatially weighted median wage 0.15 0.03 0.14 0.04
const 29.49 2.40 47.72 4.19
Vuong test 7.57 7.46
Coefficients significant at least at 5% level are in bold.∗ Coefficients significant at 10% level.
The results of the zero-inflated Poisson model are presentedin Table 6.3. The Vuong test tells
that the zero-inflated Poisson model is clearly preferred over the Poisson model. The table
presenting the estimates of the zero-inflated models consist of two parts. First part presents the
main equation, i.e. the effects of exogenous variables on the size of the commuter stream, given
that commuting is nonnegative (nonzero search intensity condition, see Section 5.3). The part of
the table with the heading ”inflate” presents the estimates of the inflation or selection equation.
The inflation equation estimates the probability that the outcome of the dependent variable is
zero. This is a discrete choice equation of zero vs. nonzero outcome, which is estimated by
logit method and provides the estimates forh1(0) in Equation 6.10. The coefficients for East
dummy for origin and destination are highly significant and negative, which supports the raw
density estimates in Figures 6.1 and 6.2 reporting lower probabilities of zero commuting for
East German regions. The effect of travel time is highly significant and positive which implies
that it is more likely that no commuting takes place between more distant regions. The estimate
72
for the median wage in the destination is highly significant and have the expected sign, meaning
that it is less likely to observe zero commuting to regions with higher wages. The estimates
for the median wage in the origin and the spread in the right tail in the origin are significant
but negative. The results imply that higher wages in the origin reduce the probability of zero
commuting. However, it does not necessarily mean that people are more likely to leave regions
with high wages. It is more likely that the negative sign is driven by the effect of wages in the
origin on participation. When wages in the origin increase,more individuals participate in the
labor market. This results in more commuting (because more people are active), however, this
should also imply that the number of stayers increases. Indeed, the results obtained by Moller
and Aldashev (2006b) show that participation rates increase with wages.
Remarkably the results of the main equation in the zero-inflated Poisson model are not much
different in magnitude from the estimates of the Poisson model. For comparison the results
of the zero-inflated Poisson model where instead of two separate parameters for spreads in the
tails, only one spread parameter (the ratio of8th to 2nd decile) are shown. The spread parameter
for the region of origin is highly significant and positive. This implies that more people leave
the region when dispersion increases. The spread parameterfor the destination is statistically
insignificant.
Table 6.4: Estimation results of the zero-inflated negative binomial model (Model 1 includes two
dispersion parameters as regressors, Model 2 only one). Dependent variable - commuting
flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
East origin 0.54 0.07 0.51 0.07
East destination 0.83 0.14 0.70 0.14
log travel time −2.65 0.02 −2.65 0.02
log median wage (origin) −0.09 0.21 −0.08 0.23
log D5/D2 (origin) 0.68∗
0.39 - -
log D8/D5 (origin) 0.25 0.61 - -
log D8/D2 (origin) - - 1.94 1.15
log median wage (destination) 3.95 0.45 4.37 0.52
log D5/D2 (destination) −1.04 0.83 - -
continued. . .
73
Table 6.4: Estimation results of the zero-inflated negative binomial model (Model 1 includes two
dispersion parameters as regressors, Model 2 only one). Dependent variable - commuting
flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
log D8/D5 (destination) 7.86 1.39 - -
log D8/D2 (destination) - - 5.81 2.33
log POP (origin) 0.99 0.03 1.00 0.03
log POP (destination) 0.97 0.05 0.98 0.06
spatially weighted POP (×100) −0.17 0.03 −0.16 0.03
spatially weighted median wage−0.12 0.01 −0.12 0.01
const −12.98 1.21 −21.72 2.36
inflate
East origin −20.89 0.65 −21.57 0.56
East destination 21.63 0.85 19.51 0.81
log travel time 0.78 0.18 1.01 0.25
log median wage (origin) 0.11 3.43 −4.00 3.88
log D5/D2 (origin) −3.07 3.70 - -
log D8/D5 (origin) −15.96 5.71 - -
log D8/D2 (origin) - - −31.48 8.63
log median wage (destination) 0.75 3.05 −0.53 2.15
log D5/D2 (destination) −7.22 5.76 - -
log D8/D5 (destination) 18.38 8.20 - -
log D8/D2 (destination) - - 5.92 13.66
log POP (origin) −1.11 0.11 −1.17 0.12
log POP (destination) −1.74 0.29 −1.70 0.27
spatially weighted POP (×100) −2.76 0.45 −2.62 0.46
continued. . .
74
Table 6.4: Estimation results of the zero-inflated negative binomial model (Model 1 includes two
dispersion parameters as regressors, Model 2 only one). Dependent variable - commuting
flows.
Model 1 Model 2
variable coef. robust coef. robust
st. er. st. er.
spatially weighted median wage 0.90 0.16 0.86 0.17
const −12.94 15.91 38.99 25.67
α 1.22 0.02 1.24 0.02
Vuong test 13.74 12.06
Coefficients significant at least at 5% level are in bold.∗ Coefficients significant at least at 10% level.
The results of the zero-inflated negative binomial model arepresented in Table 6.4. The Vuong
test tells that the zero-inflated negative binomial model isclearly preferred over the negative
binomial model. Moreover, the significance ofα implies that the zero-inflated negative binomial
model is preferred over the zero-inflated Poisson.
Unlike in the zero-inflated Poisson model, the coefficient for East dummy for the destination is
positive in the inflation equation, which implies that, other things being equal, workers are more
likely not to commute to eastern regions. On the other hand the coefficient for East dummy for
origin is positive. This would mean that workers in the East are more likely to commute, and
given the previous statement,ceteris paribus, they are more likely to go to western regions.
The effect of travel time is highly significant and positive which implies that it is more likely
that no commuting takes place between more distant regions.The same result was obtained
in the zero-inflated Poisson model. However, the magnitude is almost halved (Model 1). The
results of the main equation in the zero-inflated negative binomial model are not much different
in magnitude from the estimates of the negative binomial model. Remarkably, the effect of the
travel time, median wage in the destination and population size in both origin and destination
are highly significant in all models (Poisson, negative binomial, zero-inflated Poisson, zero-
inflated negative binomial). Moreover, the effect of the median wage in the destination is almost
identical in all variants in magnitude: a one percentage increase in wages in the destination
increases the commuter flow by 4 percent.
Finally, I allow for possible interactions between East dummy and median wage and spread
75
parameters. The results of the zero-inflated negative binomial model with interaction expansion
are provided in Table 6.5.
Table 6.5: Estimation results of the zero-inflated negative binomial model (East dummy (origin and
destination) is interacted with parameters of the wage distribution). Dependent variable -
commuting flows.
variable coef. robust
st. er.
East origin 5.09 2.16
East destination −3.30 4.05
log travel time −2.66 0.02
log median wage (origin) −0.05 0.25
log D5/D2 (origin) 0.72∗
0.44
log D8/D5 (origin) 1.56 0.70
log median wage (destination) 3.00 0.51
log D5/D2 (destination) −2.61 0.96
log D8/D5 (destination) 12.68 1.63
log POP (origin) 0.99 0.02
log POP (destination) 1.00 0.05
spatially weighted POP (×100) −0.17 0.03
spatially weighted median wage −0.12 0.01
const −10.21 1.45
inflate
East origin −10.39 35.80
East destination 101.05 76.20
log travel time 0.95 0.25
log median wage (origin) −1.08 5.16
log D5/D2 (origin) −4.37 4.78
log D8/D5 (origin) −13.78∗
7.42
continued. . .
76
Table 6.5: Estimation results of the zero-inflated negative binomial model (East dummy (origin and
destination) is interacted with parameters of the wage distribution). Dependent variable -
commuting flows.
variable coef. robust
st. er.
log median wage (destination) 24.05 15.09
log D5/D2 (destination) −240.64 113.38
log D8/D5 (destination) 185.36 121.93
log POP (origin) −1.16 0.12
log POP (destination) −1.68 0.27
spatially weighted POP (×100) −2.80 0.56
spatially weighted median wage 0.91 0.20
const −82.53 104.95
α 1.21 0.02
Vuong test 12.43
N 190 532
Coefficients significant at 5% level are in bold.∗ Coefficients significant at 10% level.
Full table with all interactions is given in the Appendix.
The coefficient for the East dummy in the origin becomes ten times greater in magnitude. The
coefficient for the East dummy in the destination is insignificant unlike in the models without
interaction expansion. This implies that workers are more likely to commute out of East German
regions. The effect of travel time remains unchanged. The same can be said about population
sizes of the origin and destination. The coefficient for the median wage in the destination
is reduced in magnitude. Because median wage was also interacted with East dummy, the
estimates show that increase in the median wage in the West German region by one percent
would result in increase in commuting to this region by 3 percent. Increase in the East German
region would result in about 4.5 percentage increase in commuter flow. Unlike the results of the
zero-inflated negative binomial model without interactionexpansion, the estimates in TableA.1
show that increase in the spread below the median in the destination would result in a reduction
of the commuter flow, which could be explained by the decreasein participation. The increase
77
in the spread in the right tail of the distribution in a West German region would attract more
commuters. However, the increase in the spread in the right tail in an East German region seems
to have no significant effect.
6.7 Conclusion
In this chapter I presented a bilocational search model where individuals have an option to com-
mute if offered a job in a region other than their place of residence. It is shown that the minimal
wage the worker is willing to accept in a ”distant” region is exactly the ”local” reservation wage
plus the travel cost. Reservation wages depend on travel cost between the regions, namely,
reservation wage in the origin decreases with the travel cost whereas it increases with the travel
cost in the destination. The amount of intensity agents are willing to invest in searching also
depends on the commuting cost. It is shown that when travel costs rise, individuals reallocate
part of their search intensity from a ”distant” region to a ”local” region. This implies that the
farther the region lies, the fewer workers are expected to commute.
The empirical results show that an increase in travel time toa destination region by 1% reduces
the number of commuters by about 2.7% on average. Furthermore, this paper justifies the use
of two dispersion parameters as regressors in modeling commuting, which has not yet been
done in the literature. The empirical results claim that increase in the spread in the left tail in
the destination lowers the commuter stream, but increase inthe spread in the right tail for West
German regions pushes it up (for East German regions no significant effect of the spread above
the median is observed). Increasing the median wage in the destination by 1% raises the number
of commuters by 3% if the destination is in West Germany and byabout 4.5% if the destination
is in the East. If median wages both in origin and destinationrise by the same amount, the
expected number of commuters still goes up, which is supported by numerical simulations. The
results also support the role of ”intervening opportunities”. The estimates show that regions
with higher wages and population are more likely to be intervening opportunities. Another
important issue which is addressed in this paper is the possibility that unemployed workers do
not search in destination if the region lies far enough. The theoretical model then treats the
positive number of commuters as conditional on searching inthe destination. This is consistent
with the observed data as about one third of pairs of regions have zero commuter flows. Methods
used in the literature to handle these kinds of problems are called ”hurdle” models. However, to
the best of my knowledge there were no theoretical models substantiating the use of ”hurdle”
models on the base of search theory. The search model presented in this paper gives theoretical
justification to the use of ”zero-inflated” or ”hurdle” models for analyzing commuter flows.
78
Chapter 7
Summary, Potential Drawbacks
and Open Questions
Despite being a young actor on the stage of economic analysis, the theory of search has proven
to be very useful in bringing more insight into the problem ofunemployment, labor force par-
ticipation, wage inequality, mobility, and many other areas of labor market analysis. Job search
theory has also found diverse empirical applications. For example, models of unemployment
duration are naturally derived from the theory of job search.17 The theory of job search is able to
give alternative explanations to long-standing problems of economic theory like unemployment
persistence and wage inequality across observationally identical workers (major contributions
were made by Mortensen and Pissarides (1994), Burdett and Mortensen (1998)). Job search the-
ory has also been successfully applied to the analysis of regional participation rates in Moller
and Aldashev (2006a).
Some empirical phenomena, like declining reservation wages, could not have been answered
within a basic search model framework. Therefore various extensions to the basic model al-
lowing for nonstationarity of search were developed. Various authors contributed to elucidation
of this phenomenon, among others Gronau (1971), Mortensen (1986), Burdett and Wishwanath
(1984), van den Berg (1990). Chapter 4.1.3 of this dissertation is devoted to dynamic aspects
of search; and as I was able to show, withdrawals from labor force are the logical outcome of
the nonstationary job search. In the model, reservation wages decline due to nonstationarity of
the arrival rate. An important aspect of the model is that wealth-maximizing agents solve not
only for optimal reservation wage but also for optimal withdrawal time, called potential search
time in this work, which is the maximum amount of time the unemployed workers are willing
to allocate to search. If an agent is unsuccessful during thepotential search time he withdraws
from the labor market (he could also be considered as a ”discouraged worker”). Moreover, the
79
model establishes a tradeoff between the probability of finding a job and the length of potential
search; hence, a change in a variable which results in reduction in the reservation wage would
on the one hand increase the worker’s employment chances fora given period of time, but on
the other hand it results in decrease in potential search time. This could result in a decline of
the overall (during the whole search span) job finding probability. This tradeoff between the
probability of finding a job and the length of potential search poses certain problems when one
tries to compare performances of different labor markets. Section 4.2 suggests the methods for
correcting the estimated job finding rate for the withdrawalprobability.
The simulation results presented in Section 3.3 show that reducing the level of unemployment
insurance benefits may even worsen the employment situationdue to a higher transition rate
into nonparticipation. The measures aimed at increasing the arrival rate directly or indirectly
seemed to be more effective. This is especially the case if the unemployment insurance benefits
are generous compared to the utility of nonparticipation toencourage the unemployed to stay
active in the labor market and participate in the active labor market programs if required.
The theory of job search has also been successfully applied to the analysis of labor mobility.
The model of locational search is presented in Chapter 5. Themodel shows that the reservation
wage is not unique to an individual, i.e. agents set different reservation wages for the ”local”
and ”distant” region. Moreover, I allow for arrival rate to be the function of search intensity
and therefore unemployed workers receive job offers only inthose regions where their search
intensity is nonzero. In this setting the maximal acceptable travel distance is determined by
the condition of nonzero search intensity. It is shown that exogenous factors in ”local” region
affect the reservation wage and search intensity in ”distant” region and vice versa. The model
shows that the circle of possible commuting destinations isbounded and the maximum distance
a worker is willing to commute is determined by exogenous factors in the model. The hurdle or
zero-inflated models which are often applied to count data containing many zeros are a natural
choice advised by the theoretical model.
It is common in the search literature to use the mean and the mean-preserving spread to char-
acterize the wage distribution. However, I claim that the mean-preserving spread is not an
appropriate measure of dispersion in case of an asymmetric wage offer distribution. In order to
control for asymmetric changes in the wage dispersion one has to abandon the concept of the
mean-preserving spread as one cannot change the spread in the tails of the distribution sepa-
rately without affecting the mean. A good solution to this problem could be using the median
as a location parameter of the distribution and the median-preserving spread in the left tail as
a scale parameter of the left tail and the median-preservingspread in the right tail as a scale
parameter of the right tail of the wage distribution. This approach was pioneered by Moller
and Aldashev (2006b). The model predicts increase in the reservation wages in response to a
80
higher median-preserving spread in the right tail and decrease in response to a higher median-
preserving spread in the left tail.
The empirical results claim that increase in the spread in the left tail in the destination lowers
the commuter stream, but increase in the spread in the right tail for West German regions pushes
it up. Increasing the median wage in the destination by 1% raises the number of commuters by
3% if the destination is in West Germany and by about 4.5% if the destination is in the East.
The empirical results also support the role of ”interveningopportunities”. The estimates show
that regions with higher wages and population are more likely to attract commuters.
There are still many challenges facing a researcher in the field of job search. Analysis of par-
ticipation behavior and withdrawals from the labor market has received so far little attention.
The empirical testing is also rather tricky so far because itis not always possible to identify
a ”discouraged worker”. Many problems facing a researcher in job search are data driven:
reservation wages are rarely observed, but even so, the measurement error of the self-reported
reservation wages is an issue. Moreover, IABS data, for example, contain information on the
unemployed receiving the unemployment benefits. Some unemployed are officially registered
only to receive the unemployment compensation but do not actively search for work. On the
other hand, some of the jobless workers are actively seekingemployment but are not entitled to
receive unemployment benefits and therefore do not appear inthe administrative data. Hence,
there is need for data where it is undoubtedly possible to identify the active job search state.
The chapter on commuting provides an alternative search model when the changes in the wage
offer distribution are asymmetric. The model derives the sign of the effects of the changes in
the spread parameters on the reservation wage unambiguously. However, the theoretical effect
on commuting is ambiguous. Therefore, the estimates of the commuting model only indirectly
support the theoretical results. To support the theoretical model directly, one needs to estimate
the structural parameters of the model, which does not seemsto be possible with the aggregate
data. To test the data empirically using the micro data requires identification of the place of
residence and the place of work in the data, which is not available as a scientific-use version
of the IABS. On the other hand, the effects of the changes in the spread parameters of the
distribution on participation are unambiguous (see Section 5.4). These results were empirically
supported by Moller and Aldashev (2006b). Another potential drawback which has not yet
been discussed in the locational search model presented in Section 5.2 is the competition among
searchers for jobs. If the number of vacancies in the destination is limited, then the inflow of
commuters would reduce the chances of getting a job, hence, reduce the arrival rate. A possible
extension to the model would be to endogenize the arrival rate. In a slightly different setting
this has already been done in Moller and Aldashev (2006a).
81
Notes
1In Hall (1979) a case with a linear indifference curve is alsopresented which results in a corner solution for
an efficient separation rate.
2Introduction of dividends into the model is not really critical to the model and they do not enter most of
the further calculations. It is assumed that unemployment compensation is financed by lump-sum taxation on
dividends.
3If at a certain wagew′ profits could be higher than at any other, all firms would set this wage and the wage
distribution would collapse to a one-point distribution.
4In the absence of unobserved heterogeneity reservation wages do not have to be necessarily observed, however,
reservation wages must be observed in order to identify the distribution of unobservables (see Frijters and van der
Klaauw (2006) and Flinn and Heckman (1982))
5Frijters and van der Klaauw (2006) also come to this conclusion.
6Blanchard (1996) finds that in tight labor markets, firms willnot discriminate against the long-term unem-
ployed, but in more depressed markets they will.
7Lancaster (see Lancaster (1979)) calls the hazard specification a ”second best” to studying reservation wage
itself.
8Exponential specification was called by Kiefer (1988) ”natural choice”.
9In case of censoring and ties the framework should be slightly modified. See Kiefer for details.
10I am indebted to Bernd Fitzenberger for this comment.
11Search theory is not the only candidate to explain interregional mobility. An interesting example of the effi-
ciency wage theory in a locational context can be found in Zenou (2002)
12Commuting to regionB involves a commuting or travel costδ , so to compare wages inA andB one needs to
subtract the commuting cost from the wages inB.
13see Mortensen (1986)
14The actual value is not critical; one could also assume that the lowest value of leisure isb = bmin.
15This is the preferred model. Alternative specifications canbe found in the Appendix
16Testing zero-inflated Poisson vs. zero-inflated negative binomial is straightforward as the two are nested.
17 It has to be noted, however, that the statistical theory behind duration models was developed long before
durations models found applications in labor market analysis. Duration models have been successfully applied in
many other sciences, for example, biology, medicine, engineering, and etc.
82
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89
Appendix A
Formulae Derivation
A.1 Formulae from Section 2.1.3
Derivation of Equation 2.21.
maxmj
φ(mj) = [1 − (1 − fj)mj ] wj/r − mj(c + a · Dij). (A.1)
From the first order condition:
∂φ(mj)
∂mj= −(1 − fj)
mj ln(1 − fj)wj/r − (c + a · Dij) = 0. (A.2)
It follows that (ignoring the subscriptj to save notation):
m =
ln
(
− c + a · Dij
ln(1 − f)w/r
)
ln(1 − f). (A.3)
Sinceln(1 − f) = −f for smallf , one obtains:
m = ln
(
−c + a · Dij
−f · w/r
)
/(−f) = −f−1 ln
(
c + a · Dij
f · w/r
)
= f−1 ln
(
f · w/r
c + a · Dij
)
(A.4)
A.2 Formulae from Section 2.2.1
Conditional distribution of wage offers above the reservation wage can be written as:
90
f(w|w > wR) =f(w)
Pr(w > wR), (A.5)
with Pr(w > wR) =∫
wR f(w)dw.
The expected wage, given that the offers exceed the reservation wage can be given as:
E(w|w > wR) =∫
wR
wq(z, w)f(w|w > wR)dw =
∫
wR
wq(z, w)f(w)
∫
wR q(z, w)f(w)dwdw =
1∫
wR q(z, w)f(w)dw
∫
wR
wq(z, w)f(w)dw =
∫
wR wq(z, w)f(w)dw∫
wR q(z, w)f(w)dw. (A.6)
Suppose that att = 0 the job offer is not accepted, but accepted in next period, i.e. att = 1,
then the present value of the returns to search in the next period are given by:
b − c +
∞∑
t=1
p(z, wR)E(w|w > wR)
(1 + r)t, (A.7)
whereb are the unemployment benefits andc is the search cost.
An agent continues the search in period 2 if unsuccessful in period 1, thus the present value of
the returns to search in period 2 is conditional upon being unsuccessful in period 1:
(
1 − p(z, wR))
[
b − c
1 + r+
∞∑
t=2
p(z, wR)E(w|w > wR)
(1 + r)t
]
. (A.8)
For period 3 in the same fashion:
(
1 − p(z, wR))2
[
b − c
(1 + r)2+
∞∑
t=3
p(z, wR)E(w|w > wR)
(1 + r)t
]
. (A.9)
To sum up the returns to search across time periods, it would be helpful to separate the two
terms in the square brackets. For the first term we get the following series:
b − c +(
1 − p(z, wR)) b − c
1 + r+
(
1 − p(z, wR))2 b − c
(1 + r)2+ ..., (A.10)
or:
(b − c)
1 +
1 − p(z, wR)
1 + r+
(
1 − p(z, wR))2
(1 + r)2+ ...
. (A.11)
91
The second in Equation A.11 is the diminishing geometric series, which sums up to:
1 +1 − p(z, wR)
1 + r+
(
1 − p(z, wR))2
(1 + r)2+ ... =
1
1 − 1 − p(z, wR)
1 + r
=1 + r
r + p(z, wR). (A.12)
Thus, the expression in Equation A.10 equals:
(b − c)1 + r
r + p(z, wR). (A.13)
The sum of expected wages across periods can be given as the following series:
[p(z, wR)E(w|w > wR)
1 + r+
p(z, wR)E(w|w > wR)
(1 + r)2+ · · ·
]
+
(
1 − p(z, wR))[p(z, wR)E(w|w > wR)
(1 + r)2+
p(z, wR)E(w|w > wR)
(1 + r)3+ · · ·
]
+
(
1 − p(z, wR))2[p(z, wR)E(w|w > wR)
(1 + r)3+
p(z, wR)E(w|w > wR)
(1 + r)4+ · · ·
]
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .(
1 − p(z, wR))∞
[· · ·].
(A.14)
This could be simplified to:
p(z, wR)E(w|w > wR)
1 + r
(
1 +1
1 + r+
1
(1 + r)2+ · · ·
)
+
1 − p(z, wR)
1 + r
(
1 +1
1 + r+
1
(1 + r)2+ · · ·
)
+
(1 − p(z, wR)
1 + r
)2(
1 +1
1 + r+
1
(1 + r)2+ · · ·
)
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. (A.15)
The term in square brackets is also a diminishing geometric progression and its sum can be
92
given as:
1 +1 − p(z, wR)
1 + r+
(
1 − p(z, wR)
1 + r
)2
+ · · · =1 + r
r + p(z, wR). (A.16)
Thus, Equation A.15 equals:
p(z, wR)E(w|w > wR)
1 + r
(
1 + r
r
) (
1 + r
r + p(z, wR)
)
=p(z, wR)E(w|w > wR)(1 + r)
r(r + p(z, wR).
(A.17)
Summing up A.17 and A.13 gives:
(b − c)(1 + r)
r + p(z, wR)+ p(z, wR) · E(w|w > wR)
1 + r
r(r + p(z, wR)). (A.18)
A.3 Formulae from Section 2.2.2
In continuous time case,t → 0, one obtains:
limτ→0
q(1, τ)
τ= lim
τ→0
e−λτλτ
τ= lim
τ→0e−λτλ = λ
limτ→0
q(m, τ)
τ= lim
τ→0
eλτ (λτ)m
m!τ= lim
τ→0
e−λτλmτm−1
m!; m > 1 → τm−1 = 0, lim
τ→0
q(m, τ)
τ= 0
limτ→0
1 − β(τ)
τ= lim
τ→0
1 − e−rτ
τ= lim
τ→0(re−rτ) = r
limτ→0
β(τ) = 1.
(A.19)
The Bellman equation in 2.28 considerably simplifies:
Ω(1 − β(τ)) = (b − c)τ + β(τ)[
∑∞
n=1 q(n, τ)∞∫
0
max [0, W (w)− Ω] g(wn)dw]
→
→ Ω(1 − β(τ))
τ= (b − c) + β(τ)
[
∞∑
n=1
q(n, τ)
τ
∞∫
0
max[0, W (w)− Ω]g(wn)dw]
.
(A.20)
A.4 Formulae from Section 2.3.2
The change in the share of workers earningw or less can be given as:
dG(w) = λF (w)u −(
δ + λ(1 − F (w)))
(1 − u)G(w). (A.21)
93
The first term denotes the share of unemployed workers (denoted byu) who find jobs (at a rate
λ) offering wagesw or less (given byF (w)). The second term denotes the employed workers
(1 − u) who become unemployed for exogenous reasons (at a rateδ) or who move to higher
paid jobs (at a rateλ(1 − F (w))). In the steady statedG(w) = 0 and hence:
G(w; F ) =δF (w)
δ + λ(1 − F (w)). (A.22)
The share of firms offering the lowest wage isF (b) = 0. A firm offering the lowest wage would
then have a profit of:
π(w; F ) =δλ(p − b)
M [δ + λ]2. (A.23)
A firm offering a wagew would have a profit of:
π(w; F ) =δλ(p − w)
M [δ + λ(1 − F (w))]2. (A.24)
Since in equilibrium all firms are equally profitable, equation A.23 and A.24 would yield:
F (w) =δ + λ
λ
[
1 −√
p − b
p − w
]
. (A.25)
Given the relationship between the offered and observed wages in Equation 2.53, one obtains:
G(w) =δ
λ
[√
p − b
p − w− 1
]
. (A.26)
By equatingG(w) in A.26 to 0 and to 1, one can find that the lowest wage in the economy isb
and the highest isp − (p − b)( δ
δ + λ
)2
.
The expected wage in the economy is:
E(w) =
∫ wH
wL
wdG(w), (A.27)
94
wherewL = b andwH = p − (p − b)( δ
δ + λ
)2
.
Integrating A.27 by parts yields:
E(w) = wG(w)∥
∥
∥
wH
b−
∫ wH
b
G(w)dw. (A.28)
The first term in A.28 is equal towH .
∫ wH
b
G(w)dw = −2(p − wH)1/2 δ
λ
√
p − b − δwH
λ+
2(p − b)1/2 δ
λ
√
p − b +δb
λ.
(A.29)
Hence,
E(w) = wH + 2(p − wH)1/2 δ
λ
√
p − b +δwH
λ−
2(p − b)1/2 δ
λ
√
p − b − δb
λ.
(A.30)
SubstitutingwH = p − (p − b)( δ
δ + λ
)2
into A.30 yields:
E(w) =δ
δ + λb +
λ
δ + λp. (A.31)
A.5 Formulae from Section 3.1
Collecting terms in Equation 3.2 yields:
Ω(t) − Ω(t + τ) = (b − c)τ + β(τ)∑∞
m=1 q(m, τ)∞∫
0
max[0, W (w)− Ω(t)]g(wm)dw+
+β(τ)∑∞
m=1 q(m, τ)Ω(t) + β(τ)q(0, τ, t)Ω(t + τ) − Ω(t + τ).
(A.32)
Note thatβ(τ) = e−rτ andq(0, τ, λ) = e−λ(τ), hence one can simplify Equation A.32:
95
Ω(t) − Ω(t + τ) = (b − c)τ + β(τ)∑∞
m=1 q(m, τ)∞∫
0
max[0, W (w)− Ω(t)]g(wm)dw
+β(τ)∑∞
m=1 q(m, τ)Ω(t) − Ω(t + τ)(
1 − e−rτe−λ(t)τ)
.
(A.33)
Moreover, in continuous time:
limτ→0
[
Ω(t + τ) − Ω(t)
τ
]
=dΩ(t)
dt;
limτ→0q(1, τ, λ)
τ= λ(t); lim
τ→0
q(m, τ, λ)
τ= 0, for m > 1
limτ→0
(
1 − e−rτe−λ(t)τ)
τ= r + λ(t).
(A.34)
Hence, dividing A.33 byτ and collecting terms yields:
−dΩ(t)
dt= b − c + λ(t)
∞∫
0
max 0, W (w) − Ω(t) dF (w)+
+λ(t)Ω(t) − Ω(t)(λ(t) + r) =
= b − c + λ(t)∞∫
0
max 0, W (w)− Ω(t) dF (w) − Ω(t)r.
(A.35)
Remembering thatΩ(t)r = wR(t) we can rewrite Equation A.35:
wR(t) = b − c +λ(t)
r
∞∫
wR(t)
(
w − wR(t))
dF (w) +dwR(t)
rdt, (A.36)
or alternatively:
dwR(t)
dt= rwR(t) − r(b − c) − λ(t)
∞∫
wR(t)
(
w − wR(t))
dF (w). (A.37)
96
A.6 Formulae from Section 4.1
The probability of becoming employed before timet can be written as:
P (T < t) =∞
∑
m=0
(λt)me−λt
m!
(
1 − F m(
wR))
. (A.38)
By the definition of the Poisson distribution,∑∞
m=0
(λt)me−λt
m!= 1, and, hence,
∑∞m=0
(λt)m
m!= eλt. One may rewrite Equation A.38 as:
P (T < t) = 1 − e−λt
∞∑
m=0
(
λtF(
wR))m
m!. (A.39)
Since∞
∑
m=0
(
λtF(
wR))m
m!= eλtF (wR) thenP (T < t) = 1 − e−λt(1−F (wR)).
A.7 Formulae from Section 5.2
The value of search function in the bilocational model is given as:
Ω =(
b − cA(θA) − cB(θB))
τ+
β(τ)
∑∞
n=1
∑∞
m=1 qA(m, τ)qB(n, τ)∞∫
0
max(W (x), Ω)dG(x)+
+∑∞
n=1 qA(n, τ)qB(0, τ)∞∫
0
max(W (x), Ω)dGA(x; n)+
+∑∞
n=1 qB(n, τ)qA(0, τ)∞∫
0
max
(
W (x) − δ
r, Ω
)
dGB(x; n) + qB(0, τ)qA(0, τ)Ω
.
(A.40)
We can rewrite A.40 as:
97
Ω =(
b − cA(θA) − cB(θB))
τ + β(τ)
∑∞
n=1
∑∞
m=1 qA(m, τ)qB(n, τ)∞∫
0
max(W (x) − Ω, 0)dG(x)+
+∑∞
n=1 qA(n, τ)qB(0, τ)∞∫
0
max(W (x) − Ω, 0)dGA(x; n)+
+∑∞
n=1 qB(n, τ)qA(0, τ)∞∫
0
max
(
W (x) − δ
r− Ω, 0
)
dGB(x; n) + qB(0, τ)qA(0, τ)Ω+
+∑∞
n=1
∑∞m=1 qA(m, τ)qB(n, τ)Ω +
∑∞n=1 qA(n, τ)qB(0, τ)Ω +
∑∞n=1 qB(n, τ)qA(0, τ)Ω
.
(A.41)
Consequently:
Ω =(
b − cA(θA) − cB(θB))
τ + β(τ)
∑∞
n=1
∑∞
m=1 qA(m, τ)qB(n, τ)∞∫
0
max(W (x) − Ω, 0)dG(x)+
+∑∞
n=1 qA(n, τ)qB(0, τ)∞∫
0
max(W (x) − Ω, 0)dGA(x; n)+
+∑∞
n=1 qB(n, τ)qA(0, τ)∞∫
0
max
(
W (x) − δ
r− Ω, 0
)
dGB(x; n) + Ω
.
(A.42)
SinceqA(m, τ)qB(m, τ) =e−λA(θA)τ (λA(θA)τ)m
m!× e−λB(θB)τ (λB(θB)τ)n
n!it is straightforward
to show thatlimτ=0qA(m, τ)qB(n, τ)
τ= 0, when bothm > 1 and n > 1. Hence, the
term1
τ
∞∑
n=1
∞∑
m=1
qA(m, τ)qB(n, τ)
∞∫
0
max(W (x) − Ω, 0)dG(x) = 0 in the limit. Moreover,
qA(n, τ)qB(0, τ) =e−λA(θA)τ (λA(θA)τ)n
n!e−λB(θB)τ and, hence,
limτ=0 qA(n, τ)qB(0, τ) = λA(θA) if n = 1 andlimτ=0 qA(n, τ)qB(0, τ) = 0 for n > 1. In the
same fashionlimτ=0 qB(n, τ)qA(0, τ) = λB(θB) if n = 1 andlimτ=0 qB(n, τ)qA(0, τ) = 0 for
n > 1.
Reservation wages are chosen to maximize the value of search. Knowing that it is easy to show
thatW (wRA) = Ω or wR
A = rΩ and similarlywRB = rΩ − δ.
Dividing Equation A.42 byτ and taking the limit atτ = 0 yields Equation in 5.4.
98
A.8 Proof of Proposition 1
The reservation wage equation for the regionA is given by:
wRA = b − cA(θA) − cB(θB) +
λA(θA)
r
∞∫
wRA
(w − wRA)dFA(w)+
+λB(θB)
r
∞∫
wRB
(w − wRB)dFB(w).
(A.43)
Taking the derivative with respect toδ yields:
∂wRA
∂δ= −∂wR
A
∂δ
λA(θA)
r
(
1 − FA
(
wRA
))
−(
∂wRA
∂δ+ 1
)
λB(θB)
r
(
1 − FB
(
wRB
))
= zinbw
=−λB(θB)
(
1 − FB
(
wRB
))
r + λA(θA) (1 − FA (wRA)) + λB(θB) (1 − FB (wR
B)).
(A.44)
It immediately follows that−1 6∂wR
A
∂δ6 0. HavingwR
A + δ = wRB one obtains:
∂wRB
∂δ=
r + λA(θA)(
1 − FA
(
wRA
))
r + λA(θA) (1 − FA (wRA)) + λB(θB (1 − FB (wR
B)), (A.45)
and therefore,0 6∂wR
B
∂δ6 1.
A.9 Proof of Proposition 2
To derive the effect of the change of the median on the reservation wage one needs to introduce
the notion of the translation of the distribution. Changingthe median holding the shape of the
distribution constant is simply a parallel shift of the distribution. If we increase the median
of the distributionF (x) by the valueµ, the resulting distribution would be a translation of the
original c.d.f. F (x). The distributionG(x) is a translation ofF (x) if G(x + µ) = F (x) and,
hence,G(x) = F (x − µ).
Given thatwRB = wR
A + δ, the reservation wage in the regionA is given as:
99
wRA = b−cA(θA)−cB(θB)+
λA(θA)
r
∫ ∞
wRA
(w−wRA)dFA(w)+
λB(θB)
r
∫ ∞
wRA
+δ
(w−wRA−δ)dFB(w).
(A.46)
Increase of the median in regionA by µ would result in a new reservation wage:
wRA(µ) = b − cA(θA) − cB(θB) +
λA(θA)
r
∫ ∞
wRA
(µ)
(w − wRA(µ))dFA(w − µ)+
λB(θB)
r
∫ ∞
wRA
(µ)+δ
(w − wRA(µ) − δ)dFB(w).
(A.47)
Subtracting A.47 from A.46 we obtain:
wRA(µ) − wR
A =
λA(θA)
r
[
∫ ∞
wRA
(µ)
(w − wRA(µ))dFA(w − µ) −
∫ ∞
wRA
(w − wRA)dFA(w)
]
+
λB(θB)
r
[
∫ ∞
wRA
(µ)+δ
(w − wRA(µ) − δ)dFB(w) −
∫ ∞
wRA
+δ
(w − wRA − δ)dFB(w)
]
(A.48)
By integration by parts one obtains:
∫ ∞
wRA
(µ)
(w − wRA(µ))dFA(w − µ) = EA(w) + µ − wR
A(µ) +
∫ wRA(µ)
0
FA(w − µ)dw (A.49)
∫ ∞
wRA
(w − wRA)dFA(w) = EA(w) − wR
A +
∫ wRA
0
FA(w)dw (A.50)
∫ ∞
wRA
(µ)+δ
(w − wRA(µ) − δ)dFB(w) = EB(w) − wR
A(µ) − δ +
∫ wRA(µ)+δ
0
FB(w)dw (A.51)
∫ ∞
wRA
+δ
(w − wRA − δ)dFB(w) = EB(w) − wR
A − δ +
∫ wRA
+δ)
0
FB(w)dw. (A.52)
Hence,
100
wRA(µ) − wR
A =
λA(θA)
r
(
µ − wRA(µ) + wR
A +
∫ wRA(µ)
0
FA(w − µ)dw −∫ wR
A
0
FA(w)dw)
+
λB(θB)
r
(
−wRA(µ) + wR
A +
∫ wRA(µ)+δ
0
FB(w)dw −∫ wR
A+δ
0
FB(w)dw)
.
(A.53)
Dividing the expression in A.53 byµ and taking the limit atµ = 0 with the help of the results
obtained in A.49 - A.52 one gets:
∂wRA(µ)
∂µ=
λA(θA)
r
(
1 − ∂wRA(µ)
∂µ(1 − FA(wR
A)) −∫ wR
A
0
fA(w)dw)
+
λB(θB)
r
(
−∂wRA(µ)
∂µ(1 − FB(wR
A + δ)))
(A.54)
Therefore:
∂wRA(µ)
∂µ=
λA(θA)(1 − FA(wRA))
r + λA(θA)(1 − FA(wRA)) + λB(θB)(1 − FB(wR
A + δ)). (A.55)
Obviously0 6∂wR
A(µ)
∂µ6 1. Knowing thatwR
A + δ = wRB one obtains
∂wRA
∂µ=
∂wRB
∂µ.
In the same fashion, increasing the median inB by µ:
∂wRB
∂µ=
λB(θB)(1 − FB(wRB))
r + λB(θB)(1 − FB(wRB)) + λA(θA)(1 − FA(wR
A)). (A.56)
Again,0 6∂wR
B
∂µ6 1.
For the effect of the spreads, assume that the reservation wage is below the median. Denote
Λ =∫ ∞
wR(w − wR)dF (w). One could rewrite:
Λ =w∫
wR
(w − wR)dF (w) +∞∫
w
(w − wR)dF (w) =
= w2−
w∫
wR
F (w)dw +∞∫
w
wdF (w)− wR.
(A.57)
Note that∂
∂w
∫ ∞
w
wdF (w) > 0. This result is intuitively clear – truncated mean increases if
you move the truncation point to the right. Moreover,∂
∂σR
∫ ∞
w
wdF (w) > 0 – truncated mean
101
increases if you increase the variance to the right of the truncation point. Hence,∂Λ
∂σR> 0.
The effect of the spread in the left tail is:∂Λ
∂σL= −
∫ w
wR
∂
∂σLF (w)dw < 0. The logic here is
straightforward – increasing the spread in the left tail moves some of the probability mass away
to the left of the reservation wage (fewer jobs become attractive). As a result, the reservation
wage declines to compensate for the loss of the probability mass.
Hence,∂wR
A
∂σAR
> 0 and∂wR
A
∂σAL
< 0. In the same fashion one obtains∂wR
B
∂σBR
> 0 and∂wR
B
∂σBL
< 0.
Given the relationshipwRA+δ = wR
B one also obtains:∂wR
B
∂σAR> 0 and
∂wRB
∂σAL< 0, and
∂wRA
∂σBR> 0
and∂wR
A
∂σBL
< 0.
A.10 Proof of Proposition 3
Differentiatec′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w) with respect toδ:
c′′A(θA)∂θA
∂δ=
λ′′A(θA)
r
∂θA
∂δ
∞∫
wRA
(w − wRA)dFA(w) − ∂wR
A
∂δ
(
1 − FA(wRA)
)λ′A(θA)
r. (A.58)
Hence,
∂θA
∂δ=
−∂wRA
∂δ
(
1 − FA(wRA)
)λ′A(θA)
r
c′′A(θA) − λ′′A(θA)
r
∞∫
wRA
(w − wRA)dFA(w)
> 0, (A.59)
since∂wR
A
∂δ< 0, c′′A(θA) > 0, andλ′′
A(θA) < 0.
For search intensity in regionB we can write:
c′B(θB) =λ′
B(θB)
r
∞∫
wRB
(w − wRB)dFB(w). (A.60)
102
And hence:
∂θB
∂δ=
−∂wRB
∂δ
(
1 − FB(wRB)
)λ′B(θB)
r
c′′B(θB) − λ′′B(θB)
r
∞∫
wRB
(w − wRB)dFB(w)
< 0. (A.61)
A.11 Proof of Proposition 4
Differentiatec′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w) with respect toΛA:
c′′A(θA)∂θA
∂ΛA=
λ′′A(θA)
r
∂θA
∂ΛAΛA +
λ′A(θA)
r, (A.62)
and
∂θA
∂ΛA=
λA(θA)
rc′′A(θA) − λ′′A(θA)ΛA
> 0. (A.63)
From∂θA
∂wA
=∂θA
∂ΛA
· ∂ΛA
∂wA
it immediately follows that∂θA
∂wA
> 0,∂θA
∂σAR
> 0, and∂θA
∂σAL
< 0.
Differentiatec′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w − wRA)dFA(w) with respect towA:
c′′B(θB)∂θB
∂wA
=λ′′
B(θB)
r
∂θB
∂wA
ΛB − ∂wRB
∂wA
(
1 − FB(wRB)
)λ′B(θB)
r, (A.64)
and hence,∂θB
∂wA= −
∂wRB
∂wA(1 − FB(wR
B))λ′B(θB)
rc′′B(θB) − λ′′B(θB)ΛB
< 0.
In the same fashion:∂θB
∂σAR
= −∂wR
B
∂σAR(1 − FB(wR
B))λ′B(θB)
rc′′B(θB) − λ′′B(θB)ΛB
< 0
and∂θB
∂σAL= −
∂wRB
∂σAL(1 − FB(wR
B))λ′B(θB)
rc′′B(θB) − λ′′B(θB)ΛB
> 0.
If we want to see how search intensities react to changes in wages in both regions simultane-
ously, we simply let the wage distributions in both regions be identical. Then increase in wages
103
in regionA would mean the same increase in regionB. Then,c′A(θA) =λ′
A(θA)
r
∫ ∞
wRA
(w −
wRA)dF (w) andc′B(θB) =
λ′B(θB)
r
∫ ∞
wRB
(w − wRB)dF (w). It is then easy to show that
∂θA
∂w> 0,
∂θA
∂σR> 0,
∂θA
∂σL< 0 and
∂θB
∂w> 0,
∂θB
∂σR> 0,
∂θB
∂σL< 0.
A.12 Data Used
The description of the IABS data set is taken from Moller andAldashev (2006a). The data
on wages and wage dispersion were calculated from IABS-REG.IABS-REG is a 2% random
sample from the employment register of the Federal Labor Office with regional information.
The data set includes all workers, salaried employees and trainees obliged to pay social security
contributions and covers more than 80% of all employment. Excluded are public servants,
minor employment and family workers (see Bender, Haas, and Klose (2000) for an extensive
description of the data). Because of legal sanctions for misreporting, the earnings information in
the data is highly reliable. Among others, IABS-REG contains variables on individual earnings
and skills. The regional information is based on the employer. For the empirical analysis
the data were restricted to full-time workers of the intermediate skill group (apprenticeship
completed without a university-type of education). All male and female workers were selected
that were employed on June 30th, 1997. For all regions the median wage and the second and
eighth decile of daily earnings were calculated.
INKAR database of the Federal Office for Building and Regional Planning contains basic geo-
graphic and demographic indicators on regional level. Regional population used in estimation
in Chapter 5 were taken from the INKAR dataset.
The data on commuting time was produced by the Institute for Regional Planning of the Uni-
versity of Dortmund (IRPUD). Using the data on daily commuters who report their travel time
from home to workplace, they calculate average travel time between each pair of regions (more
see in Spiekermann, Lemke, and Schurmann (2000) and citations therein).
104
A.13 Maximal Acceptable Commuting Distance, Simulations
Figure A.1: Effect of a change in the arrival rate on maximal acceptable commuting distance
Figure A.2: Effect of a change in the search cost on maximal acceptable commuting distance
105
Figure A.3: Effect of a change in the median wage (both in origin and destination) on maximalacceptable commuting distance
Figure A.4: Effect of a change in the travel cost on joint search intensity in both regions
106
A.14 Zero-inflated negative binomial estimation
Table A.1: Estimation results of the zero-inflated negative binomial model (East dummy (origin and
destination) is interacted with parameters of the wage distribution). Dependent variable -
commuting flows.
variable coef. robust
st. er.
East origin 5.09 2.16
East destination −3.30 4.05
log travel time −2.66 0.02
log median wage (origin) −0.05 0.25
log D5/D2 (origin) 0.72∗
0.44
log D8/D5 (origin) 1.56 0.70
log median wage (destination) 3.00 0.51
log D5/D2 (destination) −2.61 0.96
log D8/D5 (destination) 12.68 1.63
log POP (origin) 0.99 0.02
log POP (destination) 1.00 0.05
spatially weighted POP (×100) −0.17 0.03
spatially weighted median wage −0.12 0.01
log median wage (East origin) −0.69 0.43
log D5/D2 (East origin) −0.85 0.93
log D8/D5 (East origin) −4.12 1.37
log median wage (East destination) 1.55∗
0.82
log D5/D2 (East destination) 2.42 2.00
log D8/D5 (East destination) −13.53 2.70
const −10.21 1.45
inflate
continued. . .
107
Table A.1: Estimation results of the zero-inflated negative binomial model (East dummy (origin and
destination) is interacted with parameters of the wage distribution). Dependent variable -
commuting flows.
variable coef. robust
st. er.
East origin −10.39 35.80
East destination 101.05 76.20
log travel time 0.95 0.25
log median wage (origin) −1.08 5.16
log D5/D2 (origin) −4.37 4.78
log D8/D5 (origin) −13.78∗
7.42
log median wage (destination) 24.05 15.09
log D5/D2 (destination) −240.64 113.38
log D8/D5 (destination) 185.36 121.93
log POP (origin) −1.16 0.12
log POP (destination) −1.68 0.27
spatially weighted POP (×100) −2.80 0.56
spatially weighted median wage 0.91 0.20
log median wage (East origin) −3.71 7.85
log D5/D2 (East origin) 8.39 8.69
log D8/D5 (East origin) 19.90∗
13.87
log median wage (East destination) −23.63 14.53
log D5/D2 (East destination) 235.78 112.12
log D8/D5 (East destination) −184.89 120.12
const −82.53 104.95
α 1.21 0.02
Vuong test 12.43
N 190 532
108
Coefficients significant at 5% level are in bold.∗ Coefficients significant at 10% level.
A.15 Descriptive Statistics
Table A.2: Selected descriptive statistics of the data
variable mean st. dev.
travel time by car (in minutes) 259.44 114.08
median wage (in logs) 4.63 0.10
population (in thousands) 179.53 148.63
D8/D5 0.31 0.03
D5/D2 0.38 0.05
Total number of stayers : 18,226,796.
Total number of commuters : 8,948,362.
Number of zeros : 62,775.
Table A.3: Selected descriptive statistics (West and East Germany).
West East
variable mean st. dev. N mean st. dev. N
wage 107.34 7.78 327 90.66 5.91 110
population 197.40 164.32 327 126.97 63.53 110
unemployment rate 8.02 8.49 327 10.70 5.02 110
travel time (minutes) 256.80 39.52 327 267.08 49.16 110
Table A.4: Commuter flows between West and East German regions
Means Sums
West East West East
West 66.3 3.2 7, 112, 845 117, 992
East 12.3 101.7 453, 397 1, 264, 128
109
Table A.5: Pairwise correlation coefficients
1 2 3 4 5 6 7 8 9
(1) log D8/D5 (origin) 1.00
(2) log D8/D5 (destination) −0.00 1.00
(3) log travel time −0.03 −0.03 1.00
(4) East origin −0.80 0.00 0.05 1.00
(5) East destination 0.00 −0.80 0.05 −0.00 1.00
(6) log D5/D2 (origin) 0.72 −0.00 −0.01 −0.77 0.00 1.00
(7) log D5/D2 (destination) −0.00 0.72 −0.01 0.00 −0.77 −0.00 1.00
(8) log median wage (destination) −0.00 0.53 −0.06 0.00 −0.72 −0.00 0.47 1.00
(9) log median wage (origin) 0.53 −0.00 −0.06 −0.72 0.00 0.47 −0.00 −0.00 1.00
11
0
A.16 Withdrawals from the Labor Market
0.2
.4w
ithdr
awal
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure A.5: Single females, West Germany. Solid line - conditional, dashed line - unconditionalwithdrawal probability.
0.2
.4w
ithdr
awal
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure A.6: Married females, West Germany. Solid line - conditional, dashed line - unconditionalwithdrawal probability.
111
0.2
.4jo
b fin
ding
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure A.7: Single females, East Germany. Solid line - conditional, dashed line - unconditionalwithdrawal probability.
0.2
.4w
ithdr
awal
pro
babi
lity
0 200 400 600 800 1000duration in days
Figure A.8: Married females, East Germany. Solid line - conditional, dashed line - unconditionalwithdrawal probability.
112
A.17 Alternative Specifications
Table A.6: Estimation results of the Poisson model. Dependent variable - commuting flows.
variable coef. robust
st. er.
East origin 0.39 0.15
East destination 0.16 0.19
travel time −0.02 0.00
log median wage (origin) −0.77∗
0.40
log D5/D2 (origin) 3.15 0.70
log D8/D5 (origin) −4.39 1.24
log median wage (destination) 5.08 0.53
log D5/D2 (destination) −3.77 1.06
log D8/D5 (destination) 3.61 1.42
POP (origin) 1.80 0.20
POP (destination) 3.23 0.32
spatially weighted POP (×100) −0.29 0.07
spatially weighted median wage −0.05 0.04
const −10.07 4.46
R2 0.76
N 190 532
113
Table A.7: Estimation results of the negative binomial model. Dependent variable - commuting flows.
variable coef. robust
st. er.
East origin 0.52 0.10
East destination 0.67 0.10
travel time −0.02 0.00
log median wage (origin) 0.07 0.31
log D5/D2 (origin) 2.12 0.56
log D8/D5 (origin) 0.08 0.91
log median wage (destination) 4.67 0.56
log D5/D2 (destination) −0.33 0.97
log D8/D5 (destination) 7.47 1.62
POP (origin) 3.62 0.46
POP (destination) 3.23 0.32
spatially weighted POP (×100) −0.09 0.06
spatially weighted median wage −0.20 0.02
α 2.23 0.03
R2 0.23
N 190 532
Table A.8: Estimation results of the zero-inflated negative binomial model (Origin West Germany).
Dependent variable - commuting flows.
variable coef. robust
st. er.
East destination 0.10 0.20
travel time×100 −1.39 0.03
log median wage (origin) 0.17 0.37
log D5/D2 (origin) 2.48 0.66
continued. . .
114
Table A.8: Estimation results of the zero-inflated negative binomial model (Origin West Germany).
Dependent variable - commuting flows.
variable coef. robust
st. er.
log D8/D5 (origin) −0.21 1.26
log median wage (destination) 4.45 0.58
log D5/D2 (destination) −1.80∗
1.07
log D8/D5 (destination) 9.22 1.76
POP (origin) 3.05 0.43
POP (destination) 2.93 0.28
spatially weighted POP (×100) −0.08 0.05
spatially weighted median wage −0.17 0.03
inflate
East destination −0.07 0.49
travel time×100 0.50 0.04
log median wage (origin) 1.64 2.14
log D5/D2 (origin) −0.07 2.24
log D8/D5 (origin) −10.19 3.17
log median wage (destination) −1.89 2.45
log D5/D2 (destination) −1.71 3.60
log D8/D5 (destination) −5.02 5.61
POP (origin) −34.33 2.42
POP (destination) −23.86 3.29
spatially weighted POP (×100) 0.04 0.33
spatially weighted median wage 0.02 0.12
Vuong test 17.02
∗ Coefficients significant at 10% level.
115
Coefficients significant at 5% level are in bold.
Table A.9: Estimation results of the zero-inflated negative binomial model (Origin East Germany).
Dependent variable - commuting flows.
variable coef. robust
st. er.
East destination 1.37 0.31
travel time times 100 −0.01 0.00
log median wage (origin) −0.08 0.46
log D5/D2 (origin) −1.05 0.78
log D8/D5 (origin) 2.78 2.10
log median wage (destination) 3.92 0.65
log D5/D2 (destination) 1.51 1.26
log D8/D5 (destination) 2.78 2.10
POP (origin) 5.22 0.55
POP (destination) 3.34 0.50
spatially weighted POP (×100) −1.11 0.20
spatially weighted median wage 0.12 0.07
inflate
East destination −1.65 4.18
travel time 0.00 0.00
log median wage (origin) 5.17 7.94
log D5/D2 (origin) −6.25 9.44
log D8/D5 (origin) 9.43 7.38
log median wage (destination) −11.79∗
7.29
log D5/D2 (destination) −6.76 18.34
log D8/D5 (destination) −6.93 36.11
POP (origin) −25.23 5.67
POP (destination) −40.50∗
20.87
continued. . .
116
Table A.9: Estimation results of the zero-inflated negative binomial model (Origin East Germany).
Dependent variable - commuting flows.
variable coef. robust
st. er.
spatially weighted POP (×100) −0.08 2.27
spatially weighted median wage −0.02 0.86
Vuong test 7.71
∗ Coefficients significant at 10% level.
Coefficients significant at 5% level are in bold.
A.18 Software Used
The body text was compiled in MiKTeX 2.5 freeware using WinEdt v. 5.4, courtesy of WinEdt
Inc. Tables were produced using Gnumeric v. 1.7.6 freeware and converted into LaTeX. Simu-
lations and Figures 3.1, 3.2, 3.3, A.1, A.2, A.3 and A.4 were produced in Maple v. 10, courtesy
of Waterloo Maple Inc. Estimations of duration and count models and Figures 4.1, 4.2, 4.3, 4.4,
4.5, A.5, A.6, A.7 and A.8 were produced in Stata v. 9.2, courtesy of StataCorp LP.
117